V
Figure 4.8. Predicted suValue Using C* = 2.2 x 107 ft2 (Constant).
9 10
f(e) = function to represent the longitudinal slope influence in the
saltation length
G = weight of grain in air at one atmosphere of pressure
G+ = sediment load relative to flow transport capacity at the
toe of the sloping bed
g = acceleration of gravity
gs = sediment load (weight per unit time per unit width)
gse = total soil loss mass per unit width in a storm event
h = water depth, measured normal to bed surface
h^ = required water depth to have incipient grain motion on
horizontal beds
hÂ¡ = initial depth required to reach incipient motion
hm = average water flow depth
hs = h^ cos 6
hw = water depth plus loose soil depth
h1 = local water depth at distance x' from the bank
h* = overpressure head induced by the raindrop impacts over the
hydrostatic pressure head.
I = rainfall intensity
130 = rainfall intensity during the maximum measured 30minute
rainfall intensity during the rainstorm event
I* = I f = rainfall excess
K = roughness coefficient associated to Cf and Ref
KEa = rainfall kinetic energy per unit area
KEq = waterdrop kinetic energy
KEf = rainfall kinetic energy per unit area per unit time
Kf = soil erosivity factor in USLE
xix
297
Figure 5.6. Predicted $ and Estimated Error
Ranges in Data for Rainfall Intensities of 2.25 and 4.60
in./hr.
13
composed of two principal and sequential events. In the first one,
the soil particles are torn loose, detached from the ground surface
and made available for transport, which is the second event. There
fore, the erosive capacity of any agent was comprised of two indepen
dent variables of detaching capacity and transporting capacity. The
raindrop impacts and the surface flow runoff were the erosive agents
he considered in his study. Wind was also recognized as an individu
al erosive agent, but not included in Ellison's research.
Ellison's approach was based on four different conditions
(i.e., detachment and transportation of particles due to raindrop im
pacts or surface runoff) to describe the soil erosion process. The
detachment of soil particles by the erosive agents was related to the
soil properties and conservation practices available to the area
under stuc(y. Meanwhile, the transport of soil particles by the ero
sive agent was considered to be a function of the transportability of
the soil, the intensity of the transporting agent, and the quantity
of soil already detached.
The effect of slope and wind were mentioned as sources of
splash transportation in Ellison's studies. The kinetic energy of
the runoff, the slope, the surface roughness, the thickness of the
water layer, and the turbulence generated by the raindrop impacts
were mentioned as parameters for surface flow transportation. How
ever, Ellison did not develop expressions to define each of these
parameters. More work and knowledge were necessary before the fun
damental relationships could be obtained.
27
Mutchler and Hansen (1970) used data from Mutchler (1967) to
develop empirical dimensionless equations to represent the waterdrop
splash. They used the water layer depth to drop diameter ratio
(h/De) as the only parameter needed to obtain the other dependent
parameters already presented by Mutchler (1967).
Mutchler (1971) also presented relationships for the splash
droplet production by waterdrop impacts on a glass surface with a
water layer over it. Using the h/De ratio, he presented relation
ships for the number of droplets produced by one waterdrop impact,
the mean droplet diameter size of the droplet size distribution and
the standard deviation of that distribution. Discussions of how
these parameters changed with the water layer depth and the waterdrop
size or weight were also present.
Mutchler and Larson (1971) studied the amount of splash that a
waterdrop at terminal velocity could produce by impacting a water
layer over smooth glass at various water depths. They presented
empirical equations to predict the weight of water splashed which
indicated a maximum splash weight at h/De of 0.14 and 0.20 for De
equal to 5.6 irni and about 3 mm, respectively. The influence of
greater depth became relatively insignificant at a depth of about
threedrop diameters. They stated that without splash there cannot
be splash transport. At shallow water depths splash weights greater
than two times the waterdrop weight were observed. But as the water
depth increased to three waterdrop diameters or more, the splash
amount decreased and became relatively insignificant.
205
the dimensionless total sediment load representation. Values for $
were presented on Table 4.3.
4.2.8.2 Evaluation of Y'and ^
The flow intensity function, 'V1, was defined by Einstein (1950)
as
(Y Y) d
35
yR1 Sc
(4.6)
where
R1 = hydraulic radius due to grain roughness
Se = slope of the energy grade line
d35 = grain diameter corresponding to 35% of finer soil
material
Chiu assumed R1 = h for a wide rectangular channel where h is
the depth and that d35 = de. The timemean bed shear stress can also
be expressed
To = Yh se
(4.7)
Using those considerations and Shields definition for tcr (Equation
2.44), Chiu (1972) expressed v' as
r
Yh Se
Eh Tcr
^h ^h To
(4.8)
For this study, slope effects have to be considered on the cri
tical bed shear stress value. Therefore, a new definition of the
flow intensity function is presented as
78
characteristics of the surface in contact with the flowing water.
The raindrop impact can also modify the velocity profile specially
near the water surface at which the timemean velocity is signifi
cantly retarded due to raindrop impacts (Yoon, 1970).
Most of the researchers have found that this profile is loga
rithmic. However, the equations used to represent the velocity pro
file change from study to study depending on the selection of the
independent variables, measuring techniques used, and flow regime and
rainfall intensities studied.
Studies conducted at Texas A and M University during the 1960s
for shallow flows (water depth, h, between 1 in. (25.4 mm) and 6 in.
(152 mm)) with and without rainfall suggested the use of the Prandtl
von Krmn velocity profile
u ^m = ap + bpAn(y/h) (2.47)
v
where u is the local timemean velocity at a distance y from the
bottom, and ap and bp are empirical coefficients. These coefficients
are equal to 2.5 for clear water in pipes and deep channel flows with
no rainfall (an = bn = 2.5 = = 7^7, where < is the universal
value proposed by von Krmn). First, Glass and Smerdon (1967) found
that was higher than 0.40 for both cases, flows without rainfall
( k = 0.487 to 0.601) and for flows with rainfall (< = 0.495 to
0.805). They also observed that the mean velocity of the flow de
creased when the rainfall intensity was increased and the flow dis
charge was kept constant at the section of measurement.
273
10
8
4"
1.0
.8
.4
nrT
10
20
LEGEND
Rainfall
inVhr
mm/hr
1.25
32
o
2.25
57
3.65
93
4.60
117
*
*
30
40
Bed Slope, Sn (percent)
50
Figure 5.3. y 'Values for Given Bed Slope
360
The relative error of Yq can also be approximated from
Equation A.2 by dividing the equation by the definition of Yq,
Equation A.l.
AYd 3f(X) > AXj 3f(X) # AX2
YÂ¡~ 3XX fOO" 3 x2 fO(T
or
AYD 9 f (X) # xx AXj_ 3 f(X) # X2 AX2
Yq axx Too" x7~ sx2 Too "xÂ¡
326
effects on the raindrop splash characteristics and indications that
the splash effects decrease as the water depth increases.
The only applicable conclusion which can be brought from Pal
mer's study and some of Mutchlers studies is that the raindrop ef
fects on the splash erosion become negligible at water depths equiva
lent to three to five times the raindrop size diameter. For these
water depths or higher, the proposed model should predict the salta
tion length very well. For water depth smaller than these, the pro
posed model is mostly predicting the saltation length of the portion
of grains which have their saltation process in water only. The
portion of grains which travel through air during their saltation is
not known but must be inversely related to the water depth.
The mean water depths reported by Kilinc and Richardson are
definitely below these limiting depths. Consequently, some splash
erosion occurred, but was not measured and cannot be accounted for by
any other means. The results of this study indicate that the
physical approach used here can partially describe the saltation pro
cess in very shallow water. Fortunately, recent studies like Morgan
(1978), Walker et al. (1978), Singer and Walker (1983), and others
indicate that the assumption of neglecting splash erosion (like
Kilinc and Richardson assumed) and its effect on the average salta
tion length may not result in a significant error in the sediment
transport due to overland flow with rainfall. However, future
research and data on the saltation length of grains under very
shallow flow and with or without rainfall are needed in order to
fully account for all of the saltation processes which occur in
overland flow with rainfall.
108
In a recent study, Julien and Simons (1985) presented empirical
and analytical overland flow and stream sediment transport equations
which were compared to their own equation. They proposed a dimen
sionless sediment transport equation for overland flow based on fun
damental laws and some simplifying assumptions. The flow conditions
studied varied from laminar sheet flow on uniform soil surfaces to
turbulent flows in rills and gullies.
Similarly to Meyer and Monke's study, the values of the expo
nent of the bed slope (S) and the flow discharge per unit width (q)
were studied in order to consider the possible use of any of the
available sediment transport equations in soil erosion prediction.
Not surprisingly they concluded that most of the sediment transport
equations used for turbulent flow in water courses should not be
applied to rainfall erosion in laminar sheet flows. However, from
the equations studied, the EngelundHansen (1967) and Barekyan (1962)
equations were considered the best suited for predicting soil erosion
losses by overland flow. The Shields (1936), KalinskeBrown (Brown,
1950) and Yalin (1963) equations were also considered for possible
use, even though the exponent value for q was clearly too low in the
case of laminar sheet flows.
Based on empirical soil erosion equations available in the
literature the expected exponent values for S varied between 1.2 and
1.9 while the exponent values for q ranged from 1.4 to 2.4. These
variations were considered to be an indication of the complexity of
the soil erosion process. It also showed the dependence of the
150
slope increases can be explained (Rowlison and Martin, 1971). Since
most of the studies available in the literature present slope values
lower than 20%, the transport capacity usually controls most of the
cases and expressions which indicate all the time an increase in soil
loss as the bed slope increases are found applicable to the observed
data.
The maximum soil loss condition not only depends on the bed
slope but also depends on the overland flow conditions (e.g., flow
velocity, water depth and bed shear stress) and soil characteristics
(e.g., soil resistance, cover protection, and amount of material
available) which were discussed previously in this chapter.
The shape of the bed slope also affects the soil erosion and
runoff from a specific area. Studies like Onstad et al. (1967),
Young and Mutchler (1969a, 1969b) and others have shown that concave
slopes produce less soil loss than the uniform or convex slopes given
that all slope shapes have the same average slope gradient and
length. Young and Mutchler (1969a) reported that the maximum soil
displacement on the concave slopes took place in the upper onethird
(1/3) of their plot length with deposition occurring at the bottom of
the plot. Meanwhile, on the convex and uniform slopes, the maximum
displacement occurred at about threefourths (3/4) of the plot
length. Young and Mutchler (1969b) indicated that the average slope
was not a good indicator of soil erosion passing a given point except
for a uniform slope. Therefore, they indicated that soil losses form
irregular slopes depending on the slope of a short section of that
slope immediately above the point of measurement.
141
erosion as well as correction factors for water depth and presence of
canopy which reduced the splash erosion (or detachment). An addi
tional assumption that the total erosion rate [mass/(time unit
area)] was a linear sum of splash erosion rate and flow erosion rate
(or deposition rate) was used in this model (Park and Mitchell,
1982). The Yalin sediment transport equation was used to determine
the maximum transport capacity of the flow. Also the flow erosion
rate was related to the cross sectional average bed shear stress,
t0, to a power which usually ranged from 1.0 to 1.5. Other consi
derations of the model were basically the same as the original ver
sion of ANSWERS.
Dill aha and Beasley (1983) presented some modifications to the
original ANSWERS model. They incorporated a submodel for the calcu
lation of the particle size distribution of the eroded soil and also
used Yalin's equation with Shields' diagram modified by Mantz (1977)
to obtain the transport capacity. Other modifications were already
discussed when the Yalin equation was presented (Section 2.7.1).
Additional basic assumptions concerning detachment and transport
during shallow overland flow upon which the new model was based were
also summarized.
Knisel (1980) and Foster et al. (1981a) presented another com
puter model consuming less computer time than previous models (e.g.,
Li, 1979, or ANSWERS by Beasley et al., 1980) capable of computing
erosion and sediment transport at various times over a runoff event.
CREAMS, a field scale model for Chemicals, Runoff and Erosion from
Agricultural Management Systems, was developed with the specific
Table 4.4 IDENTIFICATION OF DATA POINTS AND GENERAL LEGEND FOR FIGURES IN THIS STUDY
Rainfall
Intensity*
in./hr (mm/hr)
5.7
Data
10
Points with Bed Slope*
Percent
15 20 30
40
Total Points
per Rainfall
Intensity
Legend
Point
1.25 (32)
I
V
IX
XIII
XVII
XXI
6
2.25 (57)
II
VI
X
XIV
XVIII
XXII
6
O
3.65 (93)
III
VII
XI
XV
XIX
XXIII
6
4.60 (117)
IV
VIII
XII
XVI
XX
XXIV
6
G
Total Points
per Slope
4
4
4
4
4
4
24 = Total
Data Points
*Data consist of only one data point for each rainfall and slope condition
Source: Kilinc and Richardson (1973)
= Manning's roughness coefficient
NMb = Manning's roughness coefficient for bare soil
NMc = Manning's roughness coefficient for rough, mulch or
vegetative covered soil
n = normalized velocity fluctuation
n0 = 3.09 = value of n corresponding to = Tcrs
n = limit of integration to obtain probability of erosion from
+ Area
n = limit of integration to obtain probability of erosion from
Area^
ns = number of straight lines into which the grainsize distribution
curve is divided
OMF = overland momentum flux
P = pressure
P* = overpressure due to raindrop impacts
Pf = erosion control practice in USLE
P^ = erosion control practice for intern'll areas
Pfr = erosion control practice for rill areas
p = absolute probability that a particle is eroded
Q = water flow discharge
q = water discharge per unit width
qÂ¡_ = lateral flow discharge per longitudinal unit length
qp = storm runoff peak
qs = volume of particles with size de transported per unit time
and unit bed width
R = Rei = rainfall erosivity factor in USLE
R' = hydraulic radius
R = resistance radius
xx i
B =
width of the crosssectional area of the flow
B0 =
buoyant force of a particle in a static fluid (horizontal
water surface)
b =
7.0
bk =
coefficient used in discharge per unit width equation of the
kinematic wave method
bm =
2.1 Clf
II
Q
Q
constant in velocity profile equation
br =
coefficient to relate rainfall intensity to the roughness
coefficient, K
C =
Chezy's coefficient
^0 =
constant determined by Chiu for deep water flow conditions
Cl =
p
dimensional function for the saltation length (length ~c)
c2 
constant representing initial dimensionless water depth
required to have incipient grain motion on a horizontal bed
C3 =
dimensionless constant
c4 =
dimensionless constant related to rainfall intensity
influence in water depth function of the saltation length
definition
c5 =
suvalue when v+dg/v = 1
o
cr>
ii
(1/2.3)*(slope of the su versus Â£n(v*de/v) curve)
ca =
sediment concentration near the top of the bed layer
Cc
canopy density cover factor
CD =
drag coefficient
cg =
ground density cover factor
cif =
clay fraction percent
Cm =
cropping management factor in USLE
Cmi
cropping management factor for intern'1 area
xvi
283
Rills may concentrate flows in some places thus increasing the
sediment transport and water depth. This also means that in other
parts of a section, water depths may be reduced and, the rainfall
effects severely intensified. Therefore, the soil detachment will be
less uniformly distributed as the rilling process increases. This
is, of course, a part of the rilling formation and soil erosion pro
cess itself.
The use of mean water depth neglects these effects and may only
give a lump value of the rainfall effects. This may create errors in
the proposed stochastic model. However, without a better means to
obtain the water depth, the model can give acceptable values only if
it is calibrated to those conditions. Nevertheless, the advantage of
using the mean water depth is that the computational process for the
sediment transport is significantly simplified. Also, the applica
bility of the model to field conditions may be based on a simpler
data collection procedure (i.e., no detailed microrelief elevation
maps, location of rills in area, etc.). In addition, this means less
computations and less time in order to reach a satisfactory solution.
5.2.9 Use of the Estimated Data Errors in the Evaluation
of the Coefficients
The least square method used to evaluate the coefficients can
consider the error of the data by providing a weighing factor for
each observed parameter value used in the model represented by Equa
tion 4.37. The weighing factor is inversely proportional to the
variance of every data value. These weighing factors would provide
234
because the value used in this study is the square root of 8 which
will be very close to unity. Due to that same reason the changes of
the square root of 8 along the longitudinal slope may be neglected.
Finally to maintain the value of the saltation length under no
rainfall condition, this study assumes that the rainfall effects on
the saltation length are represented by the expression
o o 1/2
f(IVt) = 1 C3(&: I Vt h ix cosze/(8 qZ)) (4.27)
where
f(IVt) = function to represent the rainfall parameters effects
on the saltation length
C3 = constant to be evaluated from experimental data
Equation 4.27 indicates that the saltation length will decrease
as the rainfall intensity increases as long as the water depth and
flow discharge are constant. As the water depth and/or the discharge
per unit width increases the effects of the raindrop impacts will be
reduced allowing the grains to have a longer saltation length, a
length that, of course, will not exceed that of the norainfall con
dition. With the correction factor given by Equation 4.25, the sal
tation length is expressed as a function of the flow conditions in
addition to the geometric conditions obtained from the bed slope
effects, and the grainsize effects.
As mentioned before, rainfall will also affect the depth re
quired to initiate the particle motion. Individual raindrop impacts
can detach soil particles at dimensionless water depths (h/de)
smaller than the one represented by (C2 cose) in Equation 4.18.
279
Table 5.6 AVERAGE RELATIVE ERROR OF $ AND Cs FOR EACH DATA SET
WITH SAME RAINFALL INTENSITY
Rainfall intensity (in./hr)
(mm/hr)
1.25
(32)
2.25
(57)
3.65
(93)
4.60
(117)
All
Number of points
6
6
6
6
24
$ Ave. relative error [%)
26.4
29.7
24.8
23.0
26.0
Cs Ave. relative error [%)
8.4
11.7
6.8
5.0
8.0
Table 5.7 AVERAGE RELATIVE ERROR
WITH SAME BED SLOPE
1 OF $
AND
Cs FOR
EACH
DATA SET
Bed slope (%)
5.7
10
15
20
30
40
All
Number of points
4
4
4
4
4
4
24.0
$ Ave. relative error (%)
23.3
27.5
24.5
25.8
28.3
26.4
26.0
C$ Ave. relative error (%)
5.3
9.5
6.5
7.7
10.3
8.4
8.0
107
Monke (1965) followed this approach, assuming also that the erosion
rate was equivalent to the sediment transport rate and that the over
land flow rate per unit width was proportional to the slope length,
Ls. Based on soil erosion data in the literature they indicated
that the suggested exponent values for S (1.0 to 2.0) and q (0.9
to 1.6) in the sediment transport equations were very similar to the
suggested exponent values for bed slope (1.2 to 1.6) and slope length
(1.35 to 1.6) for soil erosion relationships found in the literature.
Meyer and Monke also extended the approach of a critical value of t0
or q in terms of a critical bed slope or slope length before erosion
can take place and used them in empirical soil erosion relation
ships.
This study also compared soil erosion due to overland flow
alone to soil erosion due to overland flow with rainfall but with
the same total water discharge. Results indicated that rainfall
increased erosion of the finer particles but decreased the rate of
erosion of the larger particles. The explanation of this observation
was that rainfall induced runoff turbulence and splash allowed the
smaller particles to be transported downslope. Meanwhile, for the
case of the larger particles the decrease in flow velocity, due to
the raindrop impacts effects of leveling the bed surfaces and keeping
the rill channels broader and shallower, decreased the carrying capa
city of the flow. Meyer and Monke also indicated that their results
were intended to serve primarily as a basis for future studies of
more natural conditions.
327
5.5 The s,. Values
The model developed in this study is, theoretically, capable of
predicting soil erosion under many conditions which have not been
tested yet, such as the deeper flow with rainfall condition or any
flow depth with no rainfall condition. The model requires su to be
known or at least a relationship to obtain su must be known. This
last condition is the limiting condition for the general application
of the model.
As shown in Figure 4.11, the suvalues predicted by Equa
tion 4.41 agree very well with the expected suvalues for perfect
agreement. These suvalues also agree with the extrapolations made
from Yoon (1970) and Kisisel (1971) velocity fluctuations data.
The development of Equation 4.48 came from the relationship
Chiu (1972) obtained in his study. For different grainsize mater
ials which he considered, the sucurves presented similar slope and
shape. Also, the su changes were associated with changes in the
bed roughness, the bed forms and the corresponding resistance to the
flow. For most of the flow conditions, he presented the relationship
between su and v*de/v showed a positive slope straight segment (i.e.,
su increased linearly as Â£og v*de/v increased). All this is possible
because the only source of turbulence available in his deep water
conditions with no rainfall conditions came from the boundary ef
fects.
In this study, the main source of turbulence is the raindrop
impact. Nevertheless, the relationship between su and tog v*de/v
apparently holds for this case also. The difference is that the
219
Theoretically, as soon as water begins to flow there is a pro
bability, however small, that a small grain will move. Equation 3.40
indicates that as soon as there is a value for x0 there will be a
probability of erosion. t0 will be different from zero as soon as
the water starts flowing. Therefore, this study will consider that
the initial depth required to move grains is equal to the depression
storage elevation of the bed surface. That definition is valid for
horizontal or nearly horizontal slopes. By geometry, the initial
water depth, hj, (measured normal to the bed surface) and its corres
ponding depression storage, Z^, required to move grains will decrease
as the bed slope increases (see Figure 4.6). The increase of the
gravitational component in the direction of flow as the bed slope
increases may also help to reach incipient motion of the grain at a
lower water depth than the one required on a horizontal bed, but it
is assumed that Equation 4.13 should account for that effect.
The grain that has the highest probability of being moved first
from its original position is the one resting on the bed surface ra
ther than on the top layer of grains. This grain is one of the many
that are creating the depression storage. Figure 4.6 shows that con
dition for horizontal bed and for sloping bed. In both cases, the
water surface behind the grain is horizontal. The water depth, h, is
measured at the same reference point at the instant when (incipient)
motion of the grain is about to occur. From this figure, the initial
depth required to move the grain on a sloping bed surface may be
expressed as
132
Foster et al. (1977b) compared their equation to the original
USLE equation and a modified USLE version in which the rainfall
erosivity factor R was redefined in terms of runoff erosivity and
rainfall erosivity (Equation 2.62). The modified USLE showed an im
provement over the original USLE approach but still the new approach
proposed by Foster et al. (1977a) gave the best estimates. They con
cluded that the accuracy of erosion estimates from individual storms
could be significantly improved by using a soil loss equation having
separate terms for rill and interrill erosion. However, further
studies were needed in order to develop an accurate equation based on
sound erosion principles.
Smith (1976, 1977) presented an erosion model for watersheds in
which the kinematic wave equations and the sediment continuity equa
tion were the basic governing equations of the model. The model
could consider variation in rainfall pattern, infiltration capacity,
and complex watershed shapes and slopes. The watershed was simulated
by a watershed composed of small branched channels fed by an arrange
ment of nonrectangular, converging planes with distorted slopes.
The model used empirical equations for detachment rates due to rain
fall and flowing water as well as for the sediment transport capacity
which was represented by either shear stress or stream power func
tions. Detachment due to rainfall was expressed by Meyer and Wisch
meier's equation but also included a correction due to the water lay
er effects on raindrop impact based on Palmer's (1963, 1965) data.
88
and lateral direction. Each velocity component can be represented as
the sum of the local timemean value (u, v, w) plus the instantaneous
velocity fluctuation (u, v'
, W)
around that timemean
value. This
can be written as
(Longitudinal)
u =
Â¡j + u'
(2.56a)
(Normal or vertical)
v =
v + v1
(2.56b)
(Lateral)
w =
w + w1
(2.56c)
The only studies about this topic in overland flow with rainfall
found by this writer are Yoon (1970), Kisi sel (1971) and Shahabian
(1977). Yoon measured only the longitudinal velocity component (u')
over a physically smooth boundary. Kisisel measured both the longitu
dinal (u1) and normal (v1) velocity components over physically smooth
boundary and one rough boundary. Shahabian continued Kisi sel1s study
and obtained probability distributions of the longitudinal and normal
velocity components.
Yoon (1970) reported that for flow Reynold numbers below
Ref = 800, the longitudinal relative turbulence intensity (au/u ,
J p
where cru =r (u') ) at a given relative depth (y/h) increased sub
stantially throughout the total depth with increasing rainfall inten
sity. For higher flow Reynolds numbers rainfall intensity effects on
the relative velocity generally diminished toward the bed surface
where an approximately constant value was observed.
Except near the surface, Yoon also concluded that for a constant
rainfall intensity the relative turbulence intensity at a given depth
decreased as the flow Reynolds number increased. Near the water sur
face the turbulence depended on the raindrop intensity rather than
71
where
v*cr = timemean critical shear velocity
SGg = specific gravity of particles if fluid is water
v d
* s = particle Reynolds number
v
This entrainment function, E^, and the particle Reynolds
number were used to construct the wellknown Shields' diagram of
Figure 2.3. Using an analytical approach to the threshold movement
of a single particle, White (1940) confirmed the general form used by
Shields in spite of White's unfortunate omission of the hydrodynamic
lift on the individual grains. In addition, Yalin (1963) derived a
similar relationship of this diagram from a model for lift forces
rather than for shear forces. This adds more confidence to the
Shields' diagram.
More recently, Mantz (1977) presented an extension of the
Shields' diagram for the case of fine cohesionless grains (i.e.,
particle diameter in the range of 0.01 mm to 0.150 mm). The dashed
line in Figure 2.3 shows the regression line proposed by Mantz based
on his data and from the literature.
The use of Shields' diagram in overland flow with and without
rainfall was proposed by Foster and Meyer (1972a) as the method to
obtain the timemean critical shear stress of the soil particles ex
posed to overland flow. Also Kisisel et al. (1971) suggested using
Shields' diagram to obtain *cr which was required for use of the
DuBoys' equation in overland flow
REFERENCES
AbdelRazaq, A. Y., W. Viessman, and J. W. Hernandez, "A Solution to
the Surface Runoff Problem," J. of the Hydraulics Division,
ASCE, 93(Hy6), 335352, 1967.
Abramowitz, M., and I. A. Stegun (Eds.), Handbook of Mathematical
Functions with Formulas, Graphs and Mathematical Tables, Ninth
Printing, Dover Publications, Inc., New York, 1972.
Adams, J. E., D. Kirkham, and W. H. Scholtes, "Soil Erodibility and
Other Physical Properties of Some Iowa Soils," Iowa State
College, J. of Science, 32(4), 485540, 1958.
AlDurrah, M., and J. M. Bradford, "New Methods of Studying Soil
Detachment Due to Waterdrop Impact," J. of Soil Science Society
Society of America, 45, 949953, 1981.
AlDurrah, M. M., and J. M. Bradford, "Parameters for Describing Soil
Detachment Due to Single Waterdrop Impact," J. of Soil Science
of America, 46, 836840, 1982a.
AlDurrah, M. M., and J. M. Bradford, "The Mechanism of Raindrop
Splash on Soil Surfaces," J. of Soil Science Society of America,
46, 10861090, 1982b.
Alonso, C. V., W. H. Neibling, and G. R. Foster, "Estimating Sediment
Transport Capacity in Watershed Modeling," Trans, of the
American Society of Agricultural Engineers, 24(5), 12111220,
rasyim.
Barekyan, A. S., "Discharge of Channel Forming Sediments and Elements
of Sand Waves," Soviet Hydrology Selected Papers No. 2, Trans,
of the American Geophysical Union, EOS, 128130, 1962.
Barfield, B. J., "Studies of Turbulence in Shallow Sediment Laden
Flow with Superimposed Rainfall," Water Resources Institute,
Technical Report 11, Texas A and M University, 1968.
Barnett, A. P., "A Decade of Kfactor Evaluation in the Southeast,"
in Soil Erosion: Prediction and Control, Proc. of a National
Conference on Soil Erosion, May 2426, 1976, Purdue University,
West Lafayette, Indiana, Soil Conservation Society of America,
97104, 1977.
373
225
impact. Therefore, the detached particles will move farther assum
ing, of course, that they do not meet any obstruction or physical
barrier (e.g., water surface or hydraulic roughness elements) while
moving through the water layer. If the angle of impact is not normal
to the bed surface, the raindrop impacting force is decomposed in a
component normal to the bed surface and in a component tangential to
the bed surface. Consequently, the capability of the raindrop to
detach the soil particles may decrease since the normal force de
creases. Finally, the area where the raindrop makes significant con
tact with the soil bed will increase as the angle of impact departs
from 90 with respect to the bed surface. Thus the impact force is
distributed over a larger area reducing the raindrop effects per unit
area of the soil bed.
Since all of these conditions decrease the force at which soil
particles are detached it may be concluded that they will reduce the
saltation length too. All of these considerations are based on the
assumption that the flow depth and other flow conditions do not
change while the change of angle of impact is considered.
Another factor which should be taken into account is that when
the raindrop strikes the water layer, a strong downward velocity com
ponent is generated in that layer. This condition may push the
grains already in saltation back to the soil surface. Consequently,
the saltation length of such grains must be reduced. Immediately
after a raindrop strikes the soil surface an upward velocity compo
nent is generated. This upward component may push some particles
41
Other derivations for this flow condition can also be found in Chow
(1959, 1969), Grace and Eagleson (1965, 1966), Chen and Hansen
(1966), Chen and Chow (1968), Yen and Wenzel (1970), Morgali (1970),
Kilinc and Richardson (1973), and others. The presentation of terms
in the equation may change depending on the assumptions and boundary
conditions used in each study.
In general form, the onedimensional conservation of mass equa
tion for spatially varied flow for sloped beds was presented by Chen
and Chow (1968) as
+ ^ = (I f) B cos 9 + q.
at 9x L
(2.16)
The corresponding dynamic equation was presented as
9(Aum) + 9(gAum ) _
at
ax
Bj I VD B sin( 0 + fi) BL 0mqL
(2.17)
= g A sine g A S, g i[A(h cos e + h.)]
3X *
where
A = crosssectional area
f = infiltration rate
t = time
x = longitudinal distance
B = width of the channel section
fi = angle of the path of falling raindrops with respect to the
vertical axis
= momentum correction factor for the flow velocity distribu
tion
B
4.6 Initial Depth Required to Move Grain Under Very
Shallow Water Depth Conditions 220
4.7 Slope Effects on the Normal Component of the
Rainfall Momentum Flux 230
4.8 Predicted suValue Using C* = 2.2 x 107 ft^ (Constant) 245
4.9 Predicted suValue Versus Measured Sediment
Concentration using C* = 2.2 x 107 ft (Constant) . 246
4.10 Comparison of Observed and Predicted Dimensionless
Sediment Transport 252
4.11 Predicted and Required suValues 253
5.1 Bed Shear Stress Ratio 267
5.2 To_yaues as Calculated by Kilinc and Richardson (1973) 272
5.3 ^'Values for Given Bed Slope 273
5.4 ^'Values for Given Bed Slope 274
0
5.5 Observed ^Values for Given Bed Slope 277
5.6 Predicted $> and Estimated Error Ranges in Data for
Rainfall Intensities of 2.25 and 4.60 in./hr 297
5.7 Predicted $ and Estimated Error Ranges in Data for
Rainfall Intensities of 1.25 and 3.65 in./hr 298
5.8 Predicted $ Versus Observed $Values 303
5.9 Residual Values Versus the Natural Logarithm of
Observed i>Values 310
5.10 Normal Probability Plot of the Standardized Residual . 311
5.11 Saltation Length Depth Function, f(h/d0) 316
5.12 Slope Correction Factor for the Average Saltation Length. 320
5.13 Saltation Length Ratio, 2.2 x 107/C* 324
5.14 Predicted Probability of Erosion 331
5.15 Changes in p/(l p) Due to Errors in p Evaluation . 336
xi ii
SOIL EROSION BY
OVERLAND FLOW WITH RAINFALL
BY
RAUL EMILIO ZAPATA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
156
splashed particles. This means that the amount of soil transported
out of the area under study by the drop impact will be significantly
less than the amount of soil transported by overland flow. The as
sumption can also be considered valid for areas of any size with hor
izontal or nearly horizontal slopes because the splashed particles
move in all directions and the net particle transport will be zero.
As the bed slope and rainfall intensity increase the rainfallinduced
transport can be expected to also increase. However, the overland
flow will also increase and the soil transport effects can be ac
counted for in the definitions given for soil transport by overland
flow and the motion of particles by saltation.
3.2 Equilibrium Transport Condition
The overland flow and the raindrop impacts detach soil parti
cles which later on are basically transported by the overland flow.
This flow has a higher turbulence intensity due to the raindrop im
pacts. At any instant, there are particles being detached from the
ground surface while others are deposited on it. If the rate of par
ticle erosion is greater than the rate of particle deposition, the
net process will be scouring with a gradual depletion of the ground
surface elevation. When particle deposition is greater than detach
ment, the net process will be deposition with a gradual accretion of
the ground surface elevation. If the number of eroded particles cor
responds to the number of deposited particles, the ground surface
does not change elevation and it is in a state of dynamic equilibrium
usually referred to as a stable live bed surface.
330
bed slope, will make the water depth decrease while the rainfall
intensity was kept constant. So the velocity fluctuations and su
values are decreased for the previously mentioned reasons, but the
sediment discharge is increased. This is because the required bed
shear stress ratio (i.e., Tcrs/T0 = E^q') decreases as the bed slope
increases and more material will be available for motion.
Figure 5.14 presents the predicted probability of erosion for
each point used in this study, while Figure 5.1 presents the corres
ponding bed shear stress ratio of each of these points. Both of
these figures and Figure 5.3 follow practically the same pattern,
indicating the direct relationship between the bed shear stress ratio
and to the probability of erosion. It is interesting to observe
that most of the points (17 out of 24 points) have a probability of
erosion greater than 90%. This indicates how significant the rain
fall effects are on the soil erosion in very shallow overland flows.
Overland flows alone without rainfall do not have the transport
capacity to carry most of the soil particles that may be detached
from the bed surface. Therefore, the probability of erosion would be
very small (almost negligible) as the bed slope decreases because the
bed shear stress and the turbulence in the overland flow will be very
small too. This is caused by the gravitational force being the only
force available to make the water flow and thereby generate the
hydrodynamic forces (lift and drag) capable of detaching the soil
particles. For these very shallow flows the flow resistance due to
324
Rainfall
Intensity
) 5 10 15 20 25 30 35 40 45 50
Bed Slope (percent)
Figure 5.13. Saltation Length Ratio, 2.2 x 107/C,
266
modify the flow pattern over the soil surface in such manner that the
water suface is no longer parallel to the bed surface. Any flow
detention due to this obstructions or pool formations may also shift
the buoyant force toward the vertical direction. The presence of
rills may also affect the flow pattern and disrupt the ideal uniform
flow condition the buoyant force direction and the subsequent flow
condition in which Tcrs is zero.
Further research of the buoyant force effects in the incipient
motion at steep slopes is necessary. Due to the presence of high bed
shear stresses the obvious consideration could be the selection of
certain bed materials with higher critical shear stresses. For a
given grainsize, bed materials with higher specific weight (e.g.,
Barite, steel or lead balls) may help to bring close Tcrsvalues to
^values found in steep slopes. Unfortunately this will also bring
the ys/(ys y) value closer to unity. This means that the values
obtained from Equation 3.22 will approach the values from Equation
3.21 and there will not be any significant numerical difference be
tween both approaches. This is because the buoyant force direction
will no longer be the important factor to consider because the magni
tude of the weight of the particle becomes the dominant force.
Consequently, for natural soil beds on steep slopes the ^crs/^0
(see Figure 5.1 for xcrs/T0values used in this study and using
Equation 3.25 to evaluate xcrs) reduce the significance of which
203
compaction, grain shape or angularity, and grain size uniformity af
fect those angles. After reviewing some related references, it was
found in Hunt (1984) that such a wellgraded silty sand with void
ratio, en = 0.75 and dry soil density of about 94 lb/ft3 (1.5
g/cm3) could have a typical friction angle of about 32. Using
that information it was considered that an angle of repose of about
30 could be used in this study.
As it will be shown later, as the longitudinal slope increases,
the angle of repose value becomes very significant on the evaluation
of the sediment load (qs). In the absence of a better method for
the determination of a representative value of the angle of repose of
this soil, the value of 30 will be used in this study.
4.2.5 Shields Entrainment Function and Critical Bed
Shear Stress Evaluation
As it was originally presented in Figure 2.3, the Shields Dia
gram can be used to obtain the critical bed shear stress of a grain
with any given size and density. The critical bed shear stress for
horizontal beds, TCr is obtained by a trial and error proce
dure. Then the Shields entrainment function, E^ is obtained from
Equation 2.44 or directly from the diagram using the already calcu
lated critical bed shear stress of the horizontal bed.
In this study, it is necessary to correct the critical bed
shear stress for slope. Equation 3.25 is used to obtain the new
sloped critical bed shear stress, ^crs. to be used in the evalu
ation of Equation 3.41. Table 4.3 already presented the values for
Eh, xcr, and Tcrs obtained for the silty sand used in this study.
These rcr values are different from the ones reported by Kilinc and
55
an impervious surface generated by rainfall is a highly nonlinear
process and any linear analysis of the process does not strictly ap
ply. Linear models could only be used as a linear approximation of
the rainfallrunoff relationship. On the other side, the kinematic
wave model was able to better represent the rainfallrunoff rela
tionship and predicted values which agreed very well with the ob
served values. The model responded very well to changes in rainfall
intensity and slope of the runoff plane, but as observed by other
researchers, the kinematic wave model can sometimes overestimate the
discharge because of the predicted lack of the point of inflection on
the rising limb of the hydrograph.
2.5.1.2 The Law of Resistance
The DarcyWeisbach friction factor Cf is frequently used in
overland flow studies. This is expressed as
r 8gR'Sf 2%
cf 
 2
U
m
 2
P Um
(2.28)
where R' is the hydraulic radius of the crosssectional area (cross
sectional area of the flow divided by its wetted perimeter and
usually assumed equal to the flow depth, h, of the overland flow).
The friction factor is a function of the flow Reynolds number
0mh
(Ref = ^) and the boundary roughness.
For laminar flows over smooth boundaries the relationship is
(2.29)
242
lower flow Reynolds number and around 0.10 for the higher flow Rey
nolds number. That study was done on a physically smooth boundary
(glass surface) which means there were minimum roughness effects in
his results. Ki si sel1 s stuc(y considered physically smooth and rough
boundaries. His results are limited and scattered but values of su
of about 0.25 to 0.40 can be extrapolated for flow Reynolds numbers
between 500 to 1500 on the rough boundary (k = 0.0024 ft = 0.72 mm)
and rainfall intensity of 5 in./hr (127 mm/hr).
All of this seems to indicate that for low flow Reynolds num
bers the turbulence measured is due to rainfall and not due to the
boundary by itself. Since the local timemean velocity near the bed
is small, the increase in turbulence results in higher suvalues.
At high flow Reynolds numbers the turbulence measured near the bed
boundary is mainly generated by the boundary itself. The flow's rel
atively high longitudinal velocity and greater depth helps dissipa
ting the raindropgenerated turbulence before it reaches the bed sur
face. The local mean velocity near the boundary increased its value
too. Therefore, the suvalue at high flow Reynolds numbers with
rainfall approaches the typical suvalue found near the bed surface
in turbulent flow with rainfall.
As a summary, su must be expected to be a function of the
roughness (size, shape and arrangement), the particle Reynolds num
ber, the flow Reynolds number, the rainfall intensity and the water
depth. To use su in the proposed model an expression has to be
used to estimate its value. Data from Yoon (1970), Kisisel (1971),
235
The raindrop impacts are even capable of detaching particles at zero
water depth. Consequently, the water depth effects on the saltation
length need further discussion.
Equation 4.18 gives an unrealistic value (minus infinity) when
the water depth is zero. In order to have a finite value when the
water depth approaches to zero, the equation can be modified to
Co cose
f (h/de) = 1 (4.28)
(h/de) + C4 f(I)
where
f(I) = function of the rainfall properties in the water depth
function
C4 = dimensionless constant
This expression does yield a finite value at h = 0. However,
the physical consideration of this function requires a positive value
at h = 0. This is possible if the values of C2 and C4 satisfy
the relationship
C2 cose < C4 f(I) (4.29)
If this inequality is not satisfied the equation can only be
considered valid for water depths satisfying the expression
(h/de) > C2 cose C4 f(I) (4.30)
The smallest depth at which this inequality is satisfied will
be considered as the limiting depth at which the proposed model can
256
Table 5.1 ESTIMATED RELATIVE ERROR OF THE DATA
Parameter
Estimated Relative
Error (percent)*
Sediment concentration
cs
See Table 5.2
Water discharge per
unit width
q
3
Effective grain size
de
10
Bed shear stress
T
0
10
Bed slope
so
1
Angle of repose
(2730) x 100 = 6.67
Relative error = x 100
Parameter value
For this error analysis, possible errors in fluid
properties and specific weight of the soil are
presumed negligible.
22
a storm in order to describe part of a thunderstorm model. Some fre
quency distribution curves to corroborate the applicability of the
model were also suggested.
However, due to the complexity of the rainfall process many
researchers have not followed Horton's approach. Instead they have
looked for the important parameters which might affect the soil ero
sion process and have concentrated their efforts on them. In terms
of rainfall effects, researchers have studied the energy and momentum
rainfall can provide to erode the soil surface.
The kinetic energy of the rainfall can be obtained from the
raindrop size distribution for the given rainfall intensity and the
terminal velocity for each raindrop size. Based on that approach
empirical equations have been proposed. Mihara (1951) proposed the
relationship
KEt = 21,400 I1*22 (2.5)
where KEt is presented as kinetic energy per unit area and time
[erg/(cm2 min)] and I has units of mm/10 min.
Wischmeier and Smith (1958) proposed the relationship
KEa = 916 + 331 Aog(I30) (2.6)
where KE^ is the kinetic energy per unit area [(ft ton)/(acre *
in.)] and I3q is the rainfall intensity corresponding to the
maximum measured 30minute rainfall intensity during the rainstorm
event with units of in./hr. This equation is used in the Universal
43
Figure 2.1 shows a sketch defining these overland flow parame
ters. The right hand side terms of the continuity equation are the
sources of water. For the sheet flow case, the assumption of water
flowing in parallel streamlines toward the rills makes the lateral
flow term (q^) equal to zero, but for rills or any open channel flows
the lateral flow must be considered. The infiltration term is zero
if the bed surface is zero. For overland flow studies researchers
have also used the term rainfall excess, I* = rainfall intensity
minus infiltration rate. Overland flow will not exist if rainfall
excess shows a negative or zero value. This term was proposed by
Eisenlohr (1944) in a discussion of the one dimensional dynamic equa
tion derived by Keulegan (1944) for overland flow.
The terms in the momentum equation, or the socalled dynamic
equation (Equation 2.19) in order of sequence from left to right have
the following significance: (a) the unsteady term or local accel
eration term; (b) the convection acceleration term; (c) the force due
to the water weight and the friction loss term or boundary shear
force term; (d) the pressure gradient term which includes the rain
fall overpressure term; and (e) the momentum influx due to the fall
ing raindrop's component in the longitudinal slope direction. Some
studies have neglected the overpressure term but have included the
q. _
term 8(1 f + _L)Um to account for the retarding effect of rainfall
B m
excess and lateral inflow due to the mixing of the additional mass.
This term is obtained when the continuity equation is multiplied by
B0m and introduced in the momentum equation as a substitution of
the convection acceleration term.
C2 cos e
(4.32)
f(h/de) = 1 
h_ + c4 Y B; It COS2C
de (ys y)g de f(en)
Recall that the saltation length correction functions described
by Equations 4.13, 4.27, and 4.32 are to be multiplied by each other
in order to obtain the function which defines in Equation 3.10.
Hence, the average saltation length, i, may be expressed by
i = C1/de = [CQ f(e) f(h/de) f(I Vt)]/de
or
1 
tan e
tan
m
1 
Co cos e
y I Vt cos2fi
(Ys y )g de f(en)
+ C4
o
1/2
BÂ¡ I Vt h Ax cos0
8q2
>
*
(4.33)
Using the definition for Cj in this Equation 4.33 and Equa
tion 4.17 in the definition of C0, the expression for C* becomes
393
Sharma, K. D., and J. F. Correia, "An Upland Soil Erosion Model
Derived from the Basic Principles," Earth Surface Processes and
Landforms, 12, 205210, 1987.
Shen, H. S. and R. M. Li, "Rainfall Effect on Sheet Flow Over Smooth
Surface," J. of Hydraulic Division, ASCE, 99(Hy5), 771792,
1973.
Shields, A., "Anwendung der Ahnlichkeits Mechanik und Turbulenz
forschung auf die Geschiebebewegung," Mitteilungen der Preuss.
Versuchsanst fur Wasserbau und Schiffsbau, Berlin, Heft 26,
1936.
Simons, D. B., R. M. Li, and T. J. Ward, "Estimation of Sediment
Yield for a Proposed Urban Roadway Design," International
Symposium on Urban Storm Water Management, University of
Kentucky, Lexington, Kentucky, July 2427, 1978, 309315, 1978.
Simons, D. B., and F. Senturk, Sediment Transport Technology, Water
Resources Publication, Littleton, Colorado, 1977.
Singer, M. J., and J. Blanchard, "Slope Angle Interrill Soil Loss
Relationships for Slopes up to 50%," J. of Soil Science Society
of America, 46, 12701273, 1982.
Singer, M. J., and P. H. Walker, "RainfallRunoff in Soil Erosion
with Simulated Rainfall, Overland Flow and Cover," Australian J.
of Soil Research, 21, 109122, 1983.
Smart, G. M., "Sediment Transport Formula for Steep Channels," J. of
Hydraulic Engineering, ASCE, 110(Hy3), 267276, 1984.
Smerdon, E. T., "Effect of Rainfall on Critical Tractive Forces in
Channels with Shallow Flow," Trans, of the American Society of
Agricultural Engineers, 7, 287290, 1964.
Smith, D. D. and W. H. Wischmeier, "Factors Affecting Sheet and Rill
Erosion," Trans, of the American Geophysical Union, EOS, 38,
889896, IWT.
Smith, R. E., "Simulation Erosion Dynamics with a Deterministic
Distributed Watershed Model," in Proc. of the Third Federal
InterAgency Sedimentation Conference, Denver, Colorado, March
2225, 1976, Sedimentation Committee, Water Resources Council,
Ch. 1, 163173, 1976.
Smith, R. E., "Field Test of a Distributed Watershed Erosion/Sedimen
tation Model," in Soil Erosion: Prediction and Control, Proc.
of a National Conference on Soil Erosion, May 2426, 1976,
Purdue University, West Lafayette, Indiana, Soil Conservation
Society of America, 201209, 1977.
31
Wang also showed that surface tension cannot be neglected in this
kind of study.
Huang et al. (1982) neglected the body (gravity) force, the
viscosity forces and the surface tension in order to examine the
raindrop impact on a smooth rigid surface. The maximum pressure was
reported to occur at the contact circumference and that the lateral
jet velocity at the rigid surface was considered to provide the cru
cial mechanism in the raindrop soil detachment process. From this
they implied that the three critical factors important in defining
the soil resistance against the raindrop impact were the soil defor
mation characteristics, the soil shearing strength, and the soil
surface microrelief.
Then, Huang et al. (1983) presented the deformation pattern of
a solid material under a raindrop impact by numerical analysis of the
assumed linear elasticity material. The deformation due to imposed
impact loadings of: (a) a steac(y uniform load, and (b) a simulated
raindrop impact load were compared and found to be completely differ
ent. As an example, for the material with a low modulus of elasti
city, a uniform depression was found under steady, uniform load,
while a coneshaped depression was shown under the simulated raindrop
impact. As the authors indicated, this study only presented the
shape of surface deformation, but the interaction between the lateral
jet stream and the irregularities of the soil surface were believed
to be the ones which determined the amount of splashed soil.
155
final location in which these particles will reach the bed surface
again will be controlled by the same parameters as for the particles
which were transported by water alone. These parameters are the par
ticle size, its bouyant weight, its settling velocity and the water
flow conditions. Therefore, the splashed particles can be considered
to have long saltations but, of course, not as long as the ones which
represent suspension.
For this development, soil transport by overland flow is consi
dered to be the main source of soil transportation with the rainfall
sustaining the turbulence on the overland flow at higher intensities
than for overland flow without rainfall.
Soil detachment by rainfall is caused by rainfall increasing
the boundary shear stress and the intensity of turbulent velocity
fluctuations. Both phenomena help to increase the number of parti
cles being detached from the soil bed. Using this assumption, soil
detachment by rainfall and soil detachment by overland flow are con
sidered together. This new approach simplifies the approach of using
empirical relationships to define the detachment process in separate
terms which later are added together in the conservation of mass
equation. The latter approach is usually observed in the current
literature.
Soil transport by rainfall is considered to be negligible.
This assumption is valid given that the area under study is big
enough compared to the area obtained by multiplying the perimeter of
the area under study by the average horizontal distance traveled by
Table 5.3 ESTIMATED ERROR OF THE LONGITUDINAL SLOPE CORRECTION FACTOR *
Buoyant Force
Bed
Vertical
Normal to Water Surface
Between
Approaches
Slope
SC(9, )
ASC {0, <(i)
ASC (0, $)
SC(0i 4>)
ASC(0,
) ASC(e, I
Relative
Absolute
sc(0, *)
SC(0i
)
Error**
Error
(*)
(%)
(*)
(X)
Eq. 3.21
Eq. 5.10
Eq. 3.22
Eq. 5.12
0
1
0
0
1
0
0
0
0
5.7
0.8998
0.996
0.0090
0.8394
1.71
0.0144
7.20
0.0604
10
0.8227
1.91
0.0157
0.7170
3.50
0.0251
14.74
0.1057
15
0.7320
3.21
0.0235
0.5745
6.50
0.0373
27.42
0.1575
20
0.6490
4.85
0.0315
0.4327
11.4
0.0493
49.99
0.2163
30
0.4601
9.95
0.0458
0.1551
46.3
0.0718
196.6
0.3050
40
0.2852
20.8
0.0593
0
0.2852
Values
are based on
Kilinc and
Richardson's
(1973) soil
material
, angle of repose
of 30, and
1 relative
errors
of eand
5.1.
**Value obtained from Equation 3.22 assumed the true value.
Division by zero.
260
The derivative of this expression with respect to the slope
angle is
a SC (, 4>)
= sin e cos ecot
86
and the derivative of Equation 5.7 with respect to the angle of
repose is
aSC(e ,4>)
d
sin e csc2
(5.9)
By using these derivatives in Equation A.3 the estimated rela
tive error of the slope correction factor of ^ may be expressed as:
ASC(e,d>) A e(tan0 + cot4>) A0
_= _ (5.10)
SC(e,) sin2 cote sin<Â¡> cos <Â¡> 1 tane cot4> e
5.2.5 Estimated Relative Error of ^
Since Yg is generated by multiplication of Equation 4.8 by
Equation 4.9, the estimated relative error of y' is obtained from the
v
addition of Equation 5.6 to Equation 5.10 giving
Ay' Af
fg y' sinÂ£ cote sin^ cos4)
(5.11)
e(tan e+ cot$) Ae
_____
1 tan 0cot e
226
upward thus initiating or amplifying their saltation. The magnitude
of this upward velocity component is probably less than the original
downward velocity. This is because part of the energy brought into
the area under consideration by the raindrop is transferred to the
soil bed where it aids in the detachment of particles. This results
in a net timemean downward velocity component which can reduce the
saltation length of the individual grains.
To this writer's knowledge, Shahabian's (1977) study has gener
ated the only data about this phenomenon available today. That work
was done under turbulent flow conditions (Ref > 1900) with flow
depth of about 7 to 9 mm (0.023 to 0.03 ft) and over a physically
smooth surface. The general indication was that the timemean down
ward velocity component increased its magnitude as the rainfall in
tensity increased. This velocity component also increased in magni
tude as the distance from the point of measurement to the bed surface
increased. Under rainfall conditions, the standard deviation of the
vertical velocities was found to stay relatively constant at loca
tions close to bed surface and then suddenly increase in magnitude as
the location of measurement was getting closer to the water surface.
For the no rainfall condition, the trend of the velocity fluctuations
was to gradually decrease as the location of measurement moved far
off the bed surface.
The downward vertical velocities generated a net downward ver
tical momentum flux which will force the grain to have shorter salta
tion length during rainfall than without rainfall. It should be
Horizontal bed
V W
Theoretical
bed (Z=0)
FLOW*^
LWcos0
WcosOj
Wsine
Theoretical
bed
D = Drag d0 = Effective grainsize
L = Lift co =Detachment angle
W = Buoyant weight 0 = Bed slope angle
Subscript H and S represents horizontal and sloping bed conditions, respectively.
Fd = Resultant detachment force
Q = Average saltation length
Figure 3.3. Effect of Longitudinal Slope on the Saltation Length of a Grain.
Ill
types (sheet erosion, rill and gullies) were also included. Komura
also presented a simplified equation for practical applications.
Morgan (1980) derived a theoretical sediment transport equation
by combining a flow resistance formula (DarcyWeisbach friction fac
tor) with EngelundHanson (1967) formula for the sediment transport
capacity. Based on Savat's (1977) modified form of Manning's equa
tion for disturbed flow at low flow Reynolds numbers (in metric units)
.0.95
/Nh
(2.59)
where hm is the average flow depth and assuming SGs = 2.65 (the
specific gravity value of quartz), the sediment transport equation
may be written
9s
0.00611 q1,8 S1*13
j 0.15 j
nM d35
(2.60)
where d35 is the grain size corresponding to 35% of finer
materials.
This equation was compared with other earlier equations (e.g.,
Meyer and Monke, 1965; Komura, 1976 and others) and an error analysis
of the residuals was conducted for each equation. None of the equa
tions gave accurate predictions. There was an extreme variability in
sediment concentrations of the storm runoff which indicated that the
sediment transport role never reached capacity but was detachment
limited. However, the proposed equation predicted values which
compared well with those which gave better results.
95
Later, DePloey et al. (1976) showed that some of Horton's model
considerations and his noerosion belt were not valid for slope ero
sion in arid areas. They found that the infiltration capacity of a
given soil could vary as a function of rainfall intensity, slope gra
dient and runoff discharge. They also observed that the runoff dis
charge could attain a constant value at downstream locations beyond
the critical distance xcr if the infiltration capacity was equal
to the rainfall intensity. These conditions were different from the
ones Horton proposed.
They also observed that debris covers (like mulch and coarse
grains) affected the hydraulics of runoff, especially on cohesionless
soils. The debris, acting as roughness elements, may induce Itydrau
lic discontinuities (i.e., vortices) and activate local turbulent
flows which increase erosion around the debris. They especially
observed this when the flow discharge (or the flow Reynolds number)
of the sloped bed exceeded certain critical values.
DePloey et al. also reported that these large roughness ele
ments can present both slow downslope and upslope movements, which
contribute to the socalled runoff creep. The upslope movement of
particles was explained as a process in which smaller particles loca
ted below larger ones were eroded due to the locally concentrated
flows and the following collapse of the soil made up of the large
particles. Further movement depended on the soil and flow character
istics at the new location.
181
shear stress does not seem to have a significant error on qs esti
mates when the ^cr/^0 ratio on steep slopes is considered. Equation
3.21 has been frequently used by many researchers with good sediment
load estimations. It also seems to be more applicable for flow con
ditions in which Tcr/T0 ratio approaches to unity and applies for the
incipient motion at static conditions. Equation 3.22 for the uniform
flow case or Equation 3.23 as the general equation representing any
possible flow conditions appear to be more physically based but they
have not been directly tested. More studies of the orientation of
the buoyant force are necessary. Also the effects of this orienta
tion on the incipient motion criterion on steep slopes should be fur
ther explored. Based on the previous discussion and the present
stage of knowledge, it is reasonable to assume that Equation 3.21 can
be used in the present study.
Now, it is to be expected that most of the particles will be
detached toward the downstream direction but, raindrop impacts can
detach particles in any direction even upstream. The condition being
considered at this moment is the beginning of motion or detachment of
the particle from its resting location and not the direction of move
ment. Therefore, the absolute values of D and xQ are used. Sub
stituting the expressions for D, L, and W on Inequality (3.19)
yields
 tq  > (a2^a1^ de^Ys (cos e sin e cot<{>)
(X/  Xo  ) + cot
(3.24)
momentum terms due to the incoming rainfall flux. The local velocity
fluctuations are assumed to be distributed according to the Gaussian
law.
Chiu's saltation process assumed the saltation length to be
inversely proportional to the particle size and independent of the
flow conditions. In this study the effect of the longitudinal bed
slope, water depth, and rainfall parameters on the saltation length
is included. However, distance traveled in air by particles splashed
by raindrops and the number of particles which travel airborne cannot
be fully accounted for by the proposed definition.
An error analysis was conducted to the proposed model and the
assumptions made during its development. Errors on the predicted
values were found to be similar to data errors. The maximum relative
error on the predicted values was 33%, which may be considered rea
sonable for the complex process of soil erosion by overland flow with
rainfall. The proposed model may be used in the future as an initial
step toward an improved erosion model based on the physics of this
complex process.
xx ix
248
The solution required numerical restrictions in addition to the
usual mathematical requirement that the summation of the square of
the error, Z(error)^, must be a minimum. These numerical restric
tions are
 C* cannot be a negative
 suvalues of each data point must be realistic, i.e., the
suvalues must agree with Yoons (1970) and Kisisels (1971)
extrapolated data (i.e., suvalues should not exceed 0.5
or be less than 0.1).
 The probability of erosion, p, cannot be greater than one.
(Sometimes the evaluation of p using polynomial approxima
tions can exceed the value of p = 1 when n+ goes to minus
infinity, .)
Since there are only six data points for each rainfall inten
sity, it is not statistically possible to obtain optimum values of
C5 and C6 used in the su equation for each rainfall intensity curve.
In addition there is not any information about the bed roughness
which could help to establish a relationship for C5 and C6. There
fore, these coefficients will have to be considered as constants in
this study, at least until further research produces the necessary
data.
Finally, it should be indicated that the coefficient C3 and
Equation 4.27 were eliminated in the final form of the C^function.
The reasons for this omission are that the influence of C3 and
Equation 4.27 on C* was insignificant (C* changed by less than 2%
when C3 was included) and the summation of the square of the error
did not improve the solution. It should also be noted that the
d54 = 9rain size w1th 54^ of ^iner material
dgy = grain size with 57% of finer material
dÂ£ = de value used in this study = 145 pm = 4.76 x 10~4 ft
de = diameter of effective grain size
E = rate of soil loss from USLE
Eh = critical value of Shields' entrainment function
Ej = soil erosion on intern'll areas
EV = estimated coefficient value
e = base of natural logarithm
en = void ratio of the soil
F = fraction of weight of the sediment that is finer than grain
size d
Fq = resultant detachment force
Fdh = resultant detachment force for horizontal bed
Fqc; = resultant detachment force for sloping bed
Fr = resultant force at incipient motion
Fr = 0m/(gh)*/2 = Froude Number
Ftest value = statistical value used to test hypothesis
f = infiltration rate
f(en) = 0.685/(1 + en)0,415
f(h/de) = function to represent the water depth influence in the
saltation length
f(I) = function to represent the rainfall properties in the water
depth function, f(h/de)
f(IVt) = function to represent the rainfall parameter effects in
the saltation length
xv ii i
83
v*s Ts/[P(h ^max^
t$ = timemean shear stress at the water surface
V*S .
1
2 *
du
dy
y=h
The v*s expression cannot be considered as shear velocity at
1 1/2
the water surface, but a simplifying parameter with units of { T L }.
The D' expression represents the boundary condition at the water
1/2
surface with units of {L }. This equation allows a finite velocity
gradient at the water surface instead of using the not physically
valid assumption used at the lower boundary of (d/dy) = .
y=0
Yoon found that the values of < increased as the rainfall in
tensity increased but all the kvalues were found to be less than the
usual value of <= 0.4. In this study the K values were obtained
from groups of data with the same rainfall intensity and bed slope
from which good correlation coefficients (0.903 < r < 0.963) were
obtained. No flow Reynolds number effects in the calculated K values
were considered.
Generally good agreement between the measured and predicted
local timemean velocity values was obtained although the predicted
values became smaller than the measured values near the lower boun
dary because Eauation 2.49 does not apply in the viscous sublayer
region. For constant rainfall intensity, the estimated values of the
velocity gradient at the free surface resulted to be larger with
increasing flow Reynolds number. However, the velocity gradient
decreased with increasing rainfall intensity and the flow Reynolds
number was constant.
38
associated with the susceptibility to splash erosion of the soil
which, like other previous researchers have indicated, is a complex
function of the moisture content of the sand and its grainsize dis
tribution. A technique was proposed in order to obtain the mean pro
jected splash distance along the sloped plane surface.
Park et al. (1982) used dimensional analysis to model the
splash erosion of the two possible domains; the direct impact or drop
solid domain, and the dropliquidsolid domain. The dropsolid do
main was described with analytical relationships from the conserva
tion of momentum on a sloped bed. For the dropliquidsolid domain
the water layer depth effects were related to the erosion rates by an
exponential form. The results showed that the dropsolid domain is
time dependent while the drop1iquidsol id domain is independent of
time. Bed slope effects were also considered in both domains.
Recently, Riezebos and Epema (1985) presented the importance of
drop shape on the splash erosion. They found that for all test com
binations together, the introduction of the observed drop shape in
erosivity parameters only produced minor improvements in the relation
between erosivity and detachment (or transport) by splash. However,
when they used small fall heights and low fall velocities, as in many
rainfall simulators and drop tests, the prolate drops produced a
splash detachment which was two to three times higher than the one
produced by oblate drops at impact. This was partly associated with
the high splash erosion in areas below the vegetation.
52
Henderson and Wooding, 1964; Wooding, 1965a, 1965b, 1966; Woolhiser,
1969; Eagleson, 1970; Morgali, 1970; Muzik, 1974; Li, 1979; Lane and
Shirley, 1982; Crowley II, 1982; Rose et al., 1983a, and others).
This approach uses the continuity equation for unsteady spatially
varied flow (Equation 2.18) and a simplified momentum equation in
which all terms, except bed slope (S = sine) and friction slope (Sf)
are neglected. This is based on the low numerical significance of
these terms in comparison to the magnitude of S and Sp From this
it is obtained that S = Sp A relationship between flow discharge
per unit width (q) and the flow depth is established by
(2.25)
where
a^ and b^ are coefficients expressed by the following
Laminar Flow bk = 3
bk = 5/3
(using Mann
ing's equa
tion in
English units)
Turbulent Flow
equation)
where
v = kinematic viscosity
Nm = Manning's roughness coefficient
288
5.3.2 Statistical Analysis of the Estimated Coefficient Values
Table 5.10 presents the basic statistical analysis conducted
for each of the estimated values of the coefficients used in the
proposed model. The ratio of the standard error of the estimate,
SEE, to the estimated coefficient value, EV, shows how reliable the
estimate is. The estimate will improve as the standard deviation of
SEE
the error becomes smaller (i.e., decreases).
The C5 and Cg coefficients used in Equation 4.42 to obtain
SEE
su values show very small ratios of For the C5 coefficient
SEE
the ratio of is 6.1% while for the Cg coefficient the ratio is
15.4%. As presented before, the C5 corresponds to the value of
su when the grain Reynolds number is one
v*de
= 1
and the Cg corresponds to the slope of the predicting su line
(Equation 4.35) on a semi logarithmic plot. Both values were tested
for the null hypothesis, initially considering the case in which the
coefficient is equal to zero. For a 5% level of significance, the
hypotheses were rejected for both coefficients. It was also consi
dered that su is constant with a value around 0.16 or 0.31 and,
again, both hypotheses were rejected for the 5% confidence level.
Test for su = 0.16 (constant) the Student's ttest gives
Estimated Expected
Value ~ Value = 0.58 0.16 = 11 8 > t = ? no
Standard error of estimate 0.0356 (19,0.975)
399
Zingg, A. W
Loss i
"Degree and Length of Land Slope as It Affects Soil
n Runoff," Agricultural Engineering, 21(2), 5964, 1940.
r _
207
10
O
10 1
8
6
4
2
10'i
8
6
10 1
8 :
6 
4
10
3
2 I
1
1
10
1
a
LEGEND
Rainfall
in./hr
mm/hr
1.25
32
o
o
2.25
57
3.65
93
i
4.60
117
I 1II I I II
o
_l L.
_I_L
o
_i i i iiii
4 6 8 1
10
4 6 8 1
10
i i i i i
4 6 8 1
10 :
Figure 4.2. $ Versus 4'1: The Data Points.
329
If the water depth is shallow the dropinduced velocity fluctu
ations near the bed surface are significantly increased which is ex
pected to increase the suvalues. However, for very shallow flows,
such as the ones used in this study, the roughness elements may act
as a barrier to the free motion of the water. Water depth on the
order of the roughness elements size or less (relative roughness =
k/h > 1.0) will not allow the fluid to move freely and will tend to
make the rainfall effects more local. Therefore, su may decrease
even though the rainfall intensity is increased. The raindrop im
pacts may also increase the bed roughness if the overland flow is not
strong enough to eliminate the crater generated by the raindrop im
pact. Flow conditions reported by Kilinc and Richardson indicate
that the flow Reynolds number was less than 135 (Table 4.2) and that
water depths were about 6.4 times the effective grain size, de or
less. Therefore, it is expected that the bed roughness was decreas
ing the potential maximum value of su which could have occurred on
a negligible roughness surface.
The range of the data on the flow Reynolds number is not great,
which fully accounts for the drop in su due to the increase in the
flow Reynolds number. However, the general trend of the results also
reveals a small decrease in the suvalue as Ref increased and the
bed slope was kept constant. This agrees with Yoon's observations
for flow Reynolds numbers between 450 and 4000 over a low roughness
surface.
It is necessary to mention that the data used in this study
were in the supercritical range which, joined with the increase in
367
Table B.3 PREDICTED $VALUES AND THEIR ERRORS
Run
$ Predicted
Absolute
Error
Relative
Error
(percent)
Standard
ized
Residual
I
0.00586
0.00191
24.61
1.570
V
0.02548
0.00166
6.96
0.374
IX
0.03366
0.01073
24.17
1.538
XIII
0.06388
0.01173
22.49
1.127
XVII
0.09472
0.02005
26.86
1.322
XXI
0.09157
0.00026
0.28
0.016
II
0.02698
0.00267
10.98
0.579
VI
0.15324
0.03105
25.41
1.258
X
0.21953
0.02149
8.92
0.519
XIV
0.36390
0.09685
21.02
1.311
XVIII
0.83143
0.00927
1.13
0.062
XXII
1.1143
0.05353
5.05
0.274
III
0.05755
0.00520
9.92
0.526
VII
0.29769
0.00375
1.24
0.070
XI
0.56653
0.01177
2.04
0.114
XV
1.2287
0.02156
1.79
0.098
XIX
2.3763
0.54229
29.57
1.439
XXIII
3.4650
0.46872
15.64
0.808
IV
0.11719
0.00292
2.43
0.137
VIII
0.42015
0.05631
11.82
0.699
XII
1.2146
0.17126
16.42
0.845
XVI
1.9502
0.20936
9.69
0.567
XX
2.7744
0.26473
8.71
0.506
XXIV
4.0360
1.2355
23.44
1.484
Table 2.3 SOME STREAM BEDLOAD TRANSPORT FORMULAS
o
cn
Investigator
Formula
DuBoys (1879)
% kl V'o 'cr>
where
kj = function of sediment characteristics
Schoklitsch (1930)
gs = 7,000 Sf3^2 ds1//2(q qcr) (Metric Units)
Shields (1936)
gs = 10(r/(rs v))2 q sf(Y0 icr)/ds
KalinskeBrown (Brown, 1950)
9s = 10 v* d50tV(^s T)d5o)]2
Laursen (1958)
gs = 0.01 Yq l Fj(dj/h)7/6((T/Tcr ) l) f(v*/v$.)
j J J
where
_ 2 1/3
x = (p Um /58)(d50/h) = shear stress due to grain resistance
Fj = representative fraction of grainsize dj
f(v*/vs.) = Laursen's function
J
Barekyan (1962)
qs = 0.187(Y/(YS Y))Sf q((0m Gcr)/Ucr)
40
used to describe the open channel hydraulics can be used to describe
the overland flow hydraulics if additional terms are included in or
der to account for the rainfall effects, the slope effects, and the
very shallow flow conditions of overland flow.
The most frequently used hydraulic parameters associated with
overland flow with rainfall are water depth (h), mean flow velocity
(0m), or discharge flow rate per unit width (q). These parameters
have been related to the detachment and transport of soil particles
in many empirical or semiempirical approaches. Usually, the studies
are based on the correlation between the total flow discharge at the
lower end of the area under study and the total soil loss of the
area. Other studies have also considered the rate of change of water
and soil loss with respect to time. Additional parameters which have
been used in overland flow erosion studies are bottom shear stress,
pressure at the bottom of the shallow water flow, roughness of the
bed surface (with and without vegetation), longitudinal slope of the
bed surface and its longitudinal length.
Rainfall provides input of water over the area. Depending on
infiltation, this may allow the flow discharge to change as the water
flows downslope. For this reason this flow is usually called spa
tially varied flow. The theory used to describe this flow is based
on the principles of mass and momentum conservation. Keulegan (1944)
first derived the onedimensional dynamic equation for two dimension
al spatially varied flow considering the rainfall as lateral flow.
240
However, the studies have also shown that suvalues can change with
changes of the shape and arrangement of the roughness elements. Most
of the su information has come from studies using air. Studies
where su was measured in liquid fluids are very limited. Fortun
ately, it has also been shown that the magnitude of local suvalues
are similar in air and water.
Chiu's study showed that suvalues had a relationship with
effective grainsize de and the particle Reynolds number. A semi
logarithmic plot of su versus v^dg/v for each of the bed material
mixtures showed that all of the curves had similar slope and shape.
The curves were straight lines for increasing su values between
0.10 and 0.21 while the particle Reynolds number increased. After
that, the curves bent reaching a peak value of su of about 0.215
before it began to drop. The form of the curve was also associated
with the changes on bed roughness due to changes in the bed forms.
The general tendency is that su increases as the bed roughness
increases.
In overland flow, measurement of su very close to the bed
surface is practically impossible. The main reason is that the total
depth of the flow is only a few times the sizes of the instruments
usually used to measure velocity fluctuations and thereby su in
deep flow or air. Therefore, the relative distance of the point of
measurement to the total depth (y/h) will be too far from the solid
boundary to be indicative of the near boundary kinematics. A possi
ble alternative is extrapolation of the curve toward the bed surface
149
soil increased as the 0.7 power of the slope, the 2.2 power of the
rainfall intensity and were proportional to the duration of the rain.
For dry soil conditions, he reported that the initial soil moisture
content and the condition of the soil surface also affected the soil
loss. Zingg (1940) reported that the total soil loss from a sloped
area per unit width was proportional to the 1.4 power of the bed
slope and to the 1.6 power of the horizontal projection of the
slope.
Other studies which have followed were presented in this chap
ter especially in the splash erosion (Section 2.4) and overland flow
erosion studies (Section 2.5.2), and soil erosion models (Sections
2.7.2, 2.7.3). In most of the empirical approaches, the slope ef
fects are represented by either power functions or polynomial (some
times linear) expressions based on the best fit.
The general approach of these equations is that soil erosion
increases as the bed slope increases. However, there are studies
like Foster and Martin (1979), Rowlison and Martin (1971), and
others, who have indicated that there is a decrease of soil loss as
the slope increases when a certain maximum condition involving both
slope and flow is exceeded. This has been explained by the concept
that rainfall induced detachment decreases as the slope increases
because the normal force component of the incoming raindrop decreases
as the slope increases. Then considering that the minimum value
between transport capacity and detachment rate is the one which con
trols the amount of soil being eroded, the decrease in soil loss as
Table 4.2 DIMENSIONLESS PARAMETERS CALCULATED FROM KILINC AND RICHARDSON DATA
Run
Drag
Coeffi
cient
CD
Settling
Reynolds
Number
Re$
Grain
Reynolds
Number
Rede
Flow
Reynolds
Number
Ref*
Fronde
Number
Fr*
Weber
Number
we
Normal to
Longitudinal
Momentum
Ratio
Vertical
Momentum
to Weight
Rati o
I
16.49
1.67
2.760
21.76
0.779
0.019
3.113
0.03524
V
18.77
1.44
3.762
22.73
1.023
0.027
2.607
0.03526
IX
16.49
1.67
4.228
25.90
1.387
0.035
2.265
0.03524
XIII
15.26
1.82
5.153
28.61
1.710
0.043
2.026
0.03522
XVII
15.26
1.82
5.676
29.41
2.050
0.050
1.829
0.03522
XXI
15.26
1.82
6.207
29.94
2.365
0.057
1.669
0.03522
II
19.66
1.37
3.244
50.15
1.071
0.081
2.001
0.06350
VI
18.77
1.44
4.810
52.30
1.475
0.103
1.777
0.06348
X
19.66
1.37
5.056
51.73
1.783
0.119
1.639
0.06350
XIV
18.77
1.44
5.517
53.31
2.098
0.134
1.537
0.06348
XVIII
16.49
1.67
6.176
58.11
2.739
0.161
1.369
0.06343
XXII
16.49
1.67
6.558
58.35
3.179
0.179
1.259
0.06343
254
and others which might affect the suvalue, this figure indicates
that the rainfall intensity effects on su are mainly dictated by
tq = pv^. The figure also shows how the required su is expected
to change with particle Reynolds number for a given rainfall intensi
ty. The suvalues obtained from this solution are also in good
agreement with Yoon's (1970) and Kisi sel's (1971) data.
4.3Evaluation of C+ and su 214
4.3.1 Additional Considerations on the
C^Value 214
4.3.2 su Considerations 239
4.3.3 Procedure Used to Evaluate the
Coefficients 247
4.3.4 The Values of the Coefficients C2, C4,
Cg, Cg, and m 250
V. DISCUSSIONS AND MODEL VERIFICATION 255
5.1 Introduction 255
5.2 Error Analysis of the Data 255
5.2.1 Estimated Relative Error of q .... 257
5.2.2 Estimated Relative Error of 259
5.2.3 Estimated Relative Error of 'F' 259
5.2.4 Estimated Relative Error of the Slope
Correction Factor of 'Fq 259
5.2.5 Estimated Relative Error of 'i' .... 260
5.2.6 Discussion of the Error of the
Longitudinal Slope Correction Factor 261
5.2.7 Discussion of the Estimated Relative
Errors of $ and V1 268
5.2.8 Other Possible Errors 281
5.2.9 Use of the Estimated Data Errors in
Evaluation of the Coefficients .... 283
5.3 Error Analysis of the Model and the
Predicted Values 284
5.3.1 General Statistics of the Model .... 285
5.3.2 Statistical Analysis of the Estimated
Coefficient Values 288
5.3.3 Discussion of Errors of the Model
Predicted Values 296
5.3.4 Justification of the Least Squares
Approximation Method 307
5.4 The Saltation Length Process and the
C Values 313
5.5 The suValues 327
5.6 Final Remarks 345
VI. CONCLUSIONS AND RECOMMENDATIONS 352
6.1 Conclusions 352
6.2 Recommendations 355
APPENDICES 359
A GENERAL NOTES IN THE EVALUATION OF THE ABSOLUTE
ERROR AND RELATIVE ERROR OF VARIABLES 359
viii
353
local velocity fluctuations at the top of the grains. As Chiu con
sidered in his study, the velocity fluctuations are considered to be
distributed according to the normal law. The time mean boundary
shear stress is obtained from the longitudinal momentum equation,
including the additional momentum terms due to the rainfall effects.
The use of these definitions simplifies the approach and will keep
most of the complexity of the rainfall effects in the detachment and
transportation of soil particles.
(3) The flow intensity function defined by Einstein is modified
in order to consider the bed slope contribution on the transport pro
cess. This helps to describe more clearly the rainfall and bed slope
effects on the erosion process by overland flow with rainfall where
the bed slopes can be very steep. The definition for the longitu
dinal correction factor for the critical shear stress and incipient
motion on sloping beds appears to be significantly affected by the
direction of the buoyant force. However, the selection of which con
cept should be used does not seem to significantly affect the predic
tion of the sediment transport rate.
The soil erosion equation was developed based on these concepts
and generally following the Chiu procedure. Using published data
from Kilinc and Richardson (1973) the constants of the proposed model
were determined. The developed equation, with the determined con
stants, describes very well the erosion process by overland flow with
rainfall represented by the condition reported by Kilinc and Richard
son. No other data were available to further test the capabilities
of the proposed model.
44
Figure 2.1. Definition Sketch (based on Chen and Chow, 1968).
91
had a maximum Ref = 4000 while the flows tested by Richardson and
McQuivey were highly turbulent with Ref between 25,400 and 70,500.
All this indicates that the longitudinal velocity fluctuations can be
represented by a relationship between au/v* and Y+ = yv^/v if rainfall
is not present.
Kisi sel also reported that for flows with no rainfall over both
the smooth and the rough boundaries, the normal (vertical) turbulence
intensities (av/u) were of the order of 60% to 70% of the longitudinal
turbulence intensities (^u/u). For flows with rainfall over both the
smooth and the rough boundaries with ay/u values were still of the
order of 60% to 70% of u/u values close to the boundary. However,
close to the free surface, both turbulence intensities were found to
have similar values.
Kisisel also presented some values for the Reynolds stresses,
 p u'v'. These stresses were substantially higher for flows with
rainfall over both the smooth and the rough boundaries compared to
those for flows without rainfall. The distribution along the depth
was substantially altered by the rainfall, but unfortunately, data
were too disperse and scarce that no mathematical relationship was
appropriate to describe it.
There should also be mentioned that the basic discrepancies be
tween Yoon and Kisisel results were due to the flow Reynolds number in
which the measurements were taken. Yoon collected most of his data at
Refvalues lower than Kisisel1s (Ref > 1500) and, as both authors,
Y+ = yv^/v
y distance from the bed surface to a location in the water
ymax = distance,from the bed surface to the location with
maximum u
1\\ = depression storage elevation on a horizontal bed
z = vertical distance from the bottom surface
a = dimensionless energy correction factor for the velocity
distribution of the flow
ai = level of significance for tStudent test
B
Bl
B2
Bl
Bl
r
Y
Yd
Ys
AX
6
0
= dimensionless momentum flux correction factor for the velocity
distribution of the flow
= constant of particle area
= (1 + n0su)/(1 + su2)1/2
= dimensionless momentum flux correction factor for the
distribution of the raindrop terminal velocity
= dimensionless momentum flux correction factor for the lateral
flow velocity distribution
= surface tension of water
= specific weight of water
= specific dry weight of soil material including pore volume
= specific weight of soil grains
= longitudinal length increment
= thickness of the viscous sublayer
= very small number compared to unity
= very small value of SCU
= longitudinal bed surface inclination with respect to the
horizontal
xxv
60
Their statistical analysis indicated that the uncertainty in
the selection of the friction factor for the computation of flow
depth and boundary shear stress was not too sensitive and that the
error in using incorrect friction factors was not cummulative with
each step of their numerical model. Their equations were recommended
under the conditions of being used only for 126 < Ref < 12,600,
0.5 in./hr (12.5 mm/hr) < I < 17.5 in./hr (445 mm/hr), 0.005 < S0 <
0.0108 and over a physically "smooth" boundary.
Savat (1977) presented a good summary of the hydraulics of
sheet flow on physically smooth surfaces. He also discussed some
roughness conditions and presented equations for flowmean velocity,
friction factor and Manning's N^. The variation of the exponents
of the water depth and the bed slope terms in the equations due to
the flow regime (i.e., laminar, transition or turbulent) were also
discussed. His comparison with available literature suggested that
sheet flow could be either laminar or purely turbulent, but that
mixed flows prevailed on low slopes (under 5% slopes) combined with
greater depths. He also indicated that in most cases sheet flows
were supercritical, specially on steep slopes.
Savat1s equations and experiments indicated that the effect of
raindrop impacts on the DarcyWeisbach friction factor, Cf, did not
exceed 20% in the case of laminar flow on gentle slopes. He also
indicated that the rainfall influence diminished when the discharge
or the Reynolds number increased as well as when the bed slope angle
increased. Savat also used an equation for Cf in hydraulically
144
2.8 Soil Characteristics and Slope Effects in Soil Erosion
2.8.1 Soil Characteristics
Throughout the presentation of currently available soil erosion
models the effect of the soil characteristics was recognized in prac
tically every study as a soil erodibility factor which may compensate
for the variation in the soil conditions among the data studied. The
evaluation of this factor is not simple because there are many condi
tions or properties of the soil which have certain effects on the
erosion process. It is practically impossible to obtain a relation
ship which could account for all of them. Many researchers have used
regression analysis techniques to determine the most significant
characteristics which could be used to express soil resistance to
erosion due to rainfall and/or runoff.
The current knowledge of the soil erodibility factor is still
limited and the approaches used are basically empirical. Researchers
usually use the Kf factor of the USLE (Section 2.7.2) because it is
simple and well documented. However, before the USLE was proposed
during the 1960's earlier approaches to the soil erodibility factor
or its use in a soil erosion equation included Bouyoucos (1935),
Ellison (1947), Musgrave (1947, 1954), Ekern (1953), Adams et al.
(1958) and others. More recent studies were presented in the USLE
discussion (Section 2.7.2).
Rainfall can change some of the soil characteristics which
affect the soil erosion. Raindrop impacts can break down the clods
385
Lemos, P., and J. F. Lutz, "Soil Crusting and Some Factors Affecting
It," Proc. of Soil Science Society of America 21, 485491,
1957.
Li, R. M., "Water and Sediment Routing From Watersheds," in Modeling
of Rivers, H. W. Shen (Ed.), Water Resources Research,
Littleton, Colorado, 1979.
Li, R. M., D. B. Simons, and D. R. Carder, "Mathematical Modeling of
Soil Erosion by Overland Flow," in Soil Erosion: Prediction and
Control, Proc. of a National Conference on Soil Erosion, May
2426, 1976, Purdue University, West Lafayette, Indiana, Soil
Conservation Society of America, 210216, 1977.
Li, R. M., M. A. Stevens, and D. B. Simons, "Water and Sediment
Routing from Small Watersheds," in Proc. of the Third Federal
InterAgency Sedimentation Conference, Denver, Colorado, March
2225, 1976, Sedimentation Committee, Water Resources Council,
193204, 1976.
Liggett, J. A. and D. A. Wool hi ser, "Difference Solutions of the
ShallowWater Equation," J. of the Engineering Mechanics
Division, ASCE, 91(EM2), 31P7TTTW:
Lighthill, M. J., and G. B. Whitman, "On Kinematic Waves: I. Flood
Movement in Long Rivers," Proc. of the Royal Society of London,
Series A, 229, 281316, 19W.
Lyles, L, L. A. Disrud, and N. P. Woodruff, "Effects of Soil Physical
Properties, Rainfall Characteristics, and Wind Velocity on Clod
Disintegration by Simulated Rainfall," Proc. of Soil Science
Society of America, 33, 302306, 1969.
Lyles, L., J. D. Dyckerson, and N. F. Schmeidler, "Soil detachment
from Clods by Rainfall: Effects of Wind, Mulch Cover and
Initial Soil Moisture," of the American Society of
Agricultural Engineers, 17, 697700, 1974.
Lyles, L., and N. P. Woodruff, "BoundaryLayer Flow Structure:
Effects on Detachment on Noncohesive Particles," in Sedimenta
tion (H. A. Einstein Symposium), H. W. Shen (Ed.), Fort Col 1 ins,
Colorado, 1972.
Machemehl, J. L., "Sediment Transport in Shallow Subcritical Flow
Disturbed by Simulated Rainfall," Water Resources Institute,
Technical Report No. 14, Texas A and M University, 1968.
Macklin, W. C., and P. V. Hobbs, "Subsurface Phenomena and the
Splashing of Drops on Shallow Liquids," Science, 166, 107108,
1969.
275
shown increasing as the bed slope increases, but Figure 5.3 shows
that the v' values corresponding to bed slopes of 20% or higher are
decreasing toward an asymptotic value which depends on the rainfall
intensity. So, using the flow intensity function without bed slope
effects y rather than using fg may lead to the incorrect conclusion
that the maximum sediment detachment capacity has been almost reached
on the bed slopes with 20% slope or higher. Such an erroneous condi
tion would practically generate a maximum amount of sediment being
transported out of the area under study (e.g., a maximum constant $
value independent of the bed slope). All of the data points with bed
slope of 20% or higher and of the same rainfall intensity would plot
at almost the same location on the 4> versus Â¥' diagram of Figure 4.2.
But Figure 4.2 clearly shows that o continuously increases for bed
slopes of 20% or higher. The curve is almost vertical. This indi
cates that the 't'1 definition is not precise enough to predict sedi
ment transport on steeper slopes.
After the V1 is corrected by the bed slope effects and becomes
'Fq, the sediment transport rate can be presented in a better form.
Figure 5.4 shows that the slope corrected flow intensity function,
y', continuously decreases; therefore, the erosion process continu
ously increases as the slope increases (see also Figures 4.4 or 4.5).
The conditions for incipient motion are also taken into consideration
because the bed shear stress required to initiate motion decreases as
216
instead of W. Therefore, the particle may be able to have a longer
jump if the lift and drag forces are kept the same.
However, it was shown by Equation 3.25 that the critical shear
stress decreases as the bed slope increases. Consequently, the soil
particles on sloped beds may not need to be exposed to the same drag
force as on horizontal beds to initiate motion.
Recall that the lift to drag ratio is only a function of the
roughness to grainsize ratio Christensen (1972). If the drag force
is going to change with bed slope as defined by Equation 3.21, it
must be expected that the lift force required to move the soil parti
cles on a sloped bed will change by the same factor.
Ds^Pe = = cos e sin ecot4> (4.11)
Dhor. Lhor.
or
(L/D)hor. = (L/D)slope (412>
Using this reasoning, it is expected that the lift and drag
forces acting on the horizontal bed are reduced in magnitude by the
factor presented in Equation 4.11 when the jump length on sloped bed
is considered. Based on these considerations, the effect of longitu
dinal slope, f(e), will be assumed to be represented by the expres
sion
( . 1
cos e sin ecot*
m
'
! tane
cos e
tan*
(4.13)
where
214
4.3 Evaluation of C* and su
4.3.1 Additional Considerations on the C*Value
As shown in Chapter III, C* is a function of bed slope, water
depth and the rainfall intensity. C* was expressed as
where
A]_, A2 and A3 = previously defined constants
Ci = function (bed slope, water depth, rainfall parameters)
Ci comes from the definition of the average saltation length,
i. In deep water raindrop impact effects are dissipated throughout
the depth of the water flow and do not influence the bed surface.
The saltation length is proportional to the maximum elevation the
grain can reach after it is detached from the bed surface. Assuming
steady flow, this maximum elevation may be related to the grain size
using the following reasoning. As the grain size decreases the buoy
ant weight decreases with the grain size cubed. However, the lift
force on the grain will decrease with the grain size squared. Conse
quently the resulting vertical force (L Wcose) increases and the
grain will have the potential to move farther away from the bed sur
face. The lift force gradually decreases as the grain moves far from
the bed, but the drag force (also, proportional to de^) provides the
longitudinal force to move the grain downstream. This longitudinal
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325
the water flow (splash erosion). There are two reasons that restrict
the use of Equation 4.40. First, the data reported by Kilinc and
Richardson represent the amount of material transported out of the
considered area by the water flow. No provisions to measure the
splash erosion were made in that study except for the assumption that
this kind of erosion is negligible. Secondly, the saltation path of
the grains moving through the air will be controlled by the dynamic
forces between the moving wet grain and the air. No air conditions
are used in the definition of saltation length presented in this model.
The real problem is that when the falling raindrops impact the
very shallow water and then the bed surface, some grains have enough
ejecting force to overcome the buoyant weight and surface tension for
ces. These grains will separate momentarily from the flowing water
domain while they travel through the air. The ratio of airborne
grains to the total number of grains being detached from the bed sur
face due to a raindrop impact in shallow flowing water is not known.
Also not known is the average distance traveled by those grains. The
only related information available corresponds to splash erosion due
to direct raindrop impacts on the soil surface without the protection
of a water layer over it. But this information cannot be used in this
model because it requires a continuous layer of water over the soil
surface.
The study by Palmer (1963, 1965) brought some knowledge about
water depth effects on the erosion process, but unfortunately that
information cannot be used for the prediction of the saltation length
of grains. Other studies done by Mutch!er and others (i.e., Mutchler,
1967; Mutchler and Hansen, 1970; Mutchler, 1971; Mutchler and Larson,
1971, and Mutchler and Young (1975) have presented more water depth
379
Farres, P., "The Role of Time and Aggregates Size in the Crusting
Process," Earth Surface Processes and Landforms 3, 243254,
1978.
Fernandez Luque, R., and R. van Beek, "Erosion and Transport of
BedLoad Sediment," J. of Hydraulic Research, 14(2), 127144,
1976.
Fogel, M. M., L. H. Hekman, Jr., and L. Duckstein, "A Stochastic
Sediment Yield Model Using the Modified Universal Soil Loss
Equation," in Soil Erosion: Prediction and Control, Proc. of a
National Conference on Soil Erosion, May 2426, 1976, Purdue
University, West Lafayette, Indiana, Soil Conservation Society
of America, 226233, 1977.
Foster, G. R., "Modeling The Erosion Process," in Hydrologic Modeling
of Small Watersheds, ASAE Monograph No. 5, C. T. Haan, H. P.
Johnson, and D. TT Brakensiek (Eds.), American Society of
Agricultural Engineers, 1982.
Foster, G. R., L. F. Huggins, and L. D. Meyer, "A Laboratory Study of
Rill Hydraulics: I. Velocity Relationships," Trans, of the
American Society of Agricultural Engineers, 27(3), 790796,
imr.
Foster, G. R., L. F. Huggins, and L. D. Meyer, "A Laboratory Study of
Rill Hydraulics: II. Shear Stress Relationships," Trans, of
the American Society of Agricultural Engineers, 27(3), 797804,
1984b.
Foster, G. R., and L. J. Lane, "Estimating Sediment Yield From
Rangeland with CREAMS," in Proc. of the Workshop on Estimating
Erosion and Sediment Yield on Rangelands, Tucson, Arizona, March
79, 1981, U.S. Dept, of Agriculture, Agricultural Reviews and
Manuals, ARMW26, 115119, 1982
Foster, G. R., L. J. Lane, J. D. Nowlin, J. M. Laflen, and R. A.
Young, "Estimating Erosion and Sediment Yield on FieldSized
Areas," Trans, of the American Society of Agricultural
Engineers, 24(5), 12531262, 1981a.
Foster, R. L., and G. L. Martin, "Effect of Unit Weight and Slope on
Erosion," J. of the Irrigation and Drainaqe Division, ASCE,
95 (IR4) 551^551", 1969.
Foster, G. R., D. K. McCool, K. G. Renard, and W. C. Moldenhauer,
"Conversion of the Universal Soil Loss Equation to SI Metric
Units," J. of Soil and Water Conservation, 36(6), 355359,
1981b.
.60
.50
.40
;u
.30
.20
.10
LEGEND
Rainfall
inVhr
mm/hr
1.25
32
o
2.25
57
3.65
93
4.60
117
I
^ su= 0.60 0,
>
I
156 (v*d e
/v>
1
o
OBO
6<*<51.B


7 8 9 10
v d
* e
Figure B.2. Required suValues
163
Rainfall
imimmuuumu
water
Figure 3.2. Schematic Saltation Length Approach for
Overland Flow.
251
C2 = 2.3
c3 = 0
C4 = 8.8
m = 0.78
*
C5 = 0.580
C6 = 0.153
C evaluation
*
su evaluation
(4.38)
(4.39)
With these values the equations describing C* and su are
2.2 x 10
7
1 
tan0
tani>
0.78
ft'2 (4.40)
2.3 cos e
h_ + 88 BI 1 vt Y
dp 0.54 g de (Y$ y)
and
su = 0.58 0.153 *n(v+de/v)
(4.41)
Figure 4.10 presents the predicted dimensionless sediment
transport rates as expressed by Equation 4.37 with the found coeffi
cient values. There is good agreement between observed and predicted
values. The largest relative error is about 33% which can be consi
dered quite acceptable considering the uncertainties involved in this
highly complex sediment transport process.
Figure 4.11 presents the suvalues predicted from Equation 4.41
and the suvalues required in order to obtain a perfect $ agreement.
Although there are factors like the bed roughness, the water depth
296
in contact with the bed surface at all times while it keeps moving.
The longitudinal velocity distribution in a vertical axis and the
downward momentum flux due to the rainfall will not further shorten
the travel length of a rolling grain. Only the grains which are in
saltation in the fluid, and therefore separated from the bed surface,
will experience a change of their saltation length. The distance
between raindrop impacts at any instant is usually significantly
greater than the expected saltation length of a typical sand grain.
Therefore, on the average, the net effect of the raindrop impacts on
the saltation length of the grains located in the area under study
may be minimal. The data used in this study apparently supports this
later reasoning.
More discussions about the estimated values of the coefficients
are presented in the sections of this chapter in which C* and su
are directly discussed (Sections 5.4 and 5.5, respectively). Prior
to this, the statistical analysis of the model and the predicting
results must be completed.
5.3.3 Discussion of Errors of the Model Predicted Values
Figures 5.6 and 5.7 show the predicted results obtained from
the present model and how they agree with the observed data and the
corresponding estimated error zones. Each of the rectangles around
an observed point represents the estimated limits of the error of the
coordinates of that point. The limits for each point are obtained
from the estimated relative error already presented in Table 5.4.
15
rainfall parameters and soil factors needed to represent the erosion
process. He recognized the use of simulated rainfall as a tool for
obtaining a better understanding of the erosion process. However, he
emphasized the need for the control of the rainfall parameters (i.e.,
rainfall intensity, and drop size, pattern, shape and velocity) in
order to have the best representation of a natural rainfall while the
soil erosion data is collected.
Like Ekern, other authors have also discussed the use of simu
lated rainfall for soil erosion research. Among them, Meyer (1965)
and Bubenzer (1979) have presented detailed information about simula
ted rainfall conditions. The general consensus of all these studies
is that the drop size distribution, the drop velocity at impact and
the rainfall intensity are the basic parameters which need to be con
trolled and duplicated to the best possible accuracy.
In the next sections of this chapter, a review of the erosive
agents presented by Ellison (i.e., raindrop and surface runoff) are
presented in more detail.
2.3 Raindrop and Rainfall Characteristics
2.3.1 Raindrop Characteristics
It was mentioned before that the raindrop impacts are the ini
tial cause for detachment of soil particles from the bed surface;
they also provide the necessary turbulence to keep the particles in
motion in the shallow overland flows. Not all raindrops which impact
the soil surface during certain periods of time are identical. So,
it is necessary to study the raindrop characteristics in order to
21
could reach again near terminal velocities in the forested area be
cause the soil did not have any other vegetation than trees. The
trees provided with 8.5 m (27.9 ft) of free fall distance between the
base of the canopy and the soil.
Other researchers (e.g., Mihara, 1951; Hudson, 1963; Carter et
al., 1974; McGregor and Mutchler, 1977; and Park et al., 1983) have
presented raindrop sizerainfall intensity relationships different
from the one proposed by Laws and Parson. Their basic differences
are considered to be due to the geographic location, climatologic
conditions, kind of rainstorm measured, the time at which the samples
were taken during the rainstorm events, and the method used to
measure the raindrop size.
The difference in the proposed empirical relationship can be
seen by presenting some of the proposed equations:
Carter et al. (1974)
D50 = 1.63 + 1.331 0.33I2 + 0.02I3 (2.2)
McGregor and Mutchler (1976)
D5q = 2.76 + 11.40 exp(1.40I) 13.16 exp(1.17I) (2.3)
Park et al. (1983)
D50 = 0.33 I0*12 (2.4)
where Djq has units of millimeters and I has units of inches per
hour except Park et al. who used I in millimeters per hour.
It should also be mentioned that Horton (1948) proposed a sta
tistical distribution of drop sizes at different spatial locations of
33
impact of falling waterdrops with a known rainfall intensity. The
amount of soil material that has been removed from the cup after a
certain period of time is considered to be the soil loss due to
splash on that soil material. Ellison (1944) and other studies have
considered that the soil detachment at a given rainfall intensity
decreases as time increases, while Bisal (1950) and others indicate
that it is constant. The latter authors consider that the decrease
in detachment rate is due to the obstruction created by the cup's
wall as the soil surface progressively decreases with time and have
proposed correction factors for the use of the splash cup technique
(Bisal, 1950; Kinnell, 1974). Farrell et al. (1974) has also pro
posed a correction factor for the geometric parameters (i.e., size
and shape) of the soil containers used to measure splash erosion.
The splash erosion has been related to soil characteristics
(i.e., particle size distribution, presence of aggregates, organic
content, and others), the bed slope, and the rainfall characteristics
(i.e., rainfall kinetic energy (or rainfall momentum) per unit area
and time, drop size, shape, and impact velocity). From it, each
study has presented empirical equations to predict the amount of soil
splash, detached or transported from a given surface area (Ellison,
1944; Ekern and Muckenhirn, 1947; Ekern, 1950; Bisal, 1960; Bubenzer
and Jones, 1971; Quansah, 1981, and Gilley and Finkner, 1985).
The use of the rainfall kinetic energy or the rainfall momentum
in those empirical equations appears to be a preference of the au
thors. However, some of these studies have presented their
270
Table 5.5 DATA POINTS WITH POTENTIALLY LARGE ERRORS OF
U
Data Point
(and Run Number)
Slope
(percent)
Rainfall
(in./hr)
Intensity
(mm/hr)
I
5.7
1.25
(32)
IX
15.0
1.25
(32)
XII
15.0
4.60
(117)
XVI
30.0
4.60
(117)
XXIV
40.0
4.60
(117)
224
impact must decrease as the water layer thickness increases. Data
collected by Palmer (1963, 1965) showed that the maximum soil detach
ment occurs when the water layer thickness was about one waterdrop
diameter. In addition, Palmer's results showed that a depth of about
threedrop diameters or more practically eliminated the soil detach
ment. In that study, the water was stagnant which means that only
raindrop splashinduced transport occurred. This information gives
an idea about the order of magnitude of the water depth at which the
shallow waterraindrop impact is important.
For this case, the saltation length is considered to be con
trolled by the grainsize, the rainfall characteristics and the over
land flow conditions. The rainfall characteristics which are consi
dered to be important are the rainfall mass flux per unit bed area,
the raindrop size and velocity at impact, the frequency of the rain
drop impacts per unit area and time, the location of those impacts
and the direction of the falling waterdrops.
The effectiveness of the rainfall to erode the soil depends on
the force (normal to the bed surface) the waterdrops can apply on the
soil. One of the initial assumptions of this model is that the wa
terdrops fall vertically and that no wind effects are considered.
However, the bed slope will not always be horizontal, in other words,
the impact will not always be normal to the bed surface.
In a perfectly perpendicular impact all of the impacting rain
drop force is fully exerted on the soil particles and a maximum ejec
ting force is obtained on the particles surrounding the point of
396
Williams, J. R., and H. D. Berndt, "Sediment Yield Prediction Based
on Watershed Hydrology," Trans, of the American Society of
Agricultural Engineers, 20, 11001104, 1977.
Wilson, B. N., B. J. Barfield, and I. D. Moore, "A Hydrology and
Sedimentology Watershed Model: Part I. Modeling Techniques,"
Department of Agricultural Engineering, University of Kentucky,
Lexington, Kentucky, 1981.
Wischmeier, W. H., "A Rainfall Erosion Index for a Universal Soil
Loss Equation," Proc. of Soil Science Society of America, 23,
246249, 1959.
Wischmeier, W. H., "CroppingManagement Factor Evaluations for a
Universal SoilLoss Equation," Proc. of Soil Science Society of
America, 24(4), 322326, 1960.
Wischmeier, W. H., "Estimating the Soil Loss Equation's Cover and
Management Factor for Undisturbed Areas," in Proc. of the
SedimentYield Workshop, USDA Sedimentation Laboratory, Oxford,
Mississippi, Nov. 2830, 1972, U. S. Dept, of Agriculture,
Agricultural Research Service ARSS40, 118124, 1975.
Wischmeier, W. H., "Use and Misuse of the Universal Soil Loss
Equation," J. of Soil and Water Conservation, 31(1), 59, 1976.
Wischmeier, W. H., C. B. Johnson, and B. V. Cross, "A Soil Erodibil
ity Nomograph for Farmland and Construction Sites," J. of Soil
and Water Conservation, 26, 189193, 1971.
Wischmeier, W. H., and J. V. Mannering, "Relation of Soil Properties
to Its Erodibility," Proc. of Soil Science Society of America,
33, 131137, 1969.
Wischmeier, W. H., and D. D. Smith, "Rainfall Energy and Its Rela
tionship to Soil Loss," Trans, of the American Geophysical
Union, 39(2), 285291, 1958.
Wischmeier, W. H., and D. D. Smith, "Predicting RainfallErosion
Losses from Cropland East of the Rocky Mountains," U. S.
Department of Agriculture, Agricultural Handbook No. 282, 1965.
Wischmeier, W. H., and D. D. Smith, "Predicting Rainfall Erosion
LossesA Guide to Conservation Planning," U. S. Department of
Agriculture, Agricultural Handbook No. 537, 1978.
Wischmeier, W. H., D. D. Smith, and R. E. Uhland, "Evaluation of
Factors in the Soil Loss Equation," Agricultural Engineering,
39, 458462, 474, 1958.
146
Farres (1978) studied soil crusting due to rainfall and formu
lated a model of the mechanism of crust development. From it the
mean crust thickness could be obtained based on the size of the soil
aggregates and the amount of water applied to the soil.
Bradford et al. (1986) studied the surface seal formation and
subsequent drying effects on the amount of soil detached by single
waterdrop impacts. They found that the splash could be correlated to
the soil strength for both uncrusted and crusted conditions for
finertexture soil but for the sands and sandy loams additional stud
ies were required.
The soil particles are not always in a cohesionless condition
and individually separated into their individual sizes. Most of the
time, as the clay and silt content increases, the grains are found in
groups or aggregates which can form clods. The raindrop impacts can
breakdown these clods and then allow the individual soil grains to be
eroded by the overland flow with rainfall. Moldenhauer and Koswara
(1968), Lyles et al. (1969, 1974) and others have studied the clod
erosion and how it is affected by: clod size and density, rainfall
intensity and duration, wind velocity, presence of mulch cover over
the clods, and initial soil moisture.
Moldenhauer and Koswara (1968) indicated that, for their soil
type studied, sheet erosion increased as the initial clod size in
creased even though the runoff decreased. In addition, the splashed
material was always coarser than the material eroded by the runoff.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iv
LIST OF TABLES x
LIST OF FIGURES xii
LIST OF SYMBOLS xv
ABSTRACT xxviii
CHAPTERS
I. INTRODUCTION 1
1.1 The Soil Erosion Problem 1
1.2 Purpose and Scope of This Study 5
II. SOIL EROSION PROCESS AND REVIEW OF
RELATED STUDIES 7
2.1 The Soil Erosion Process 7
2.2 Initial Studies 11
2.3 Raindrop and Rainfall Characteristics .... 15
2.3.1 Raindrop Characteristics 15
2.3.2 Rainfall Characteristics 19
2.4 Splash Erosion 25
2.4.1 Waterdrop Splash 25
2.4.2 Splash Erosion Studies 32
2.5 Overland Flow Erosion 39
2.5.1 Hydraulics of Overland Flow 39
2.5.1.1 Simplified Solutions, the
Kinematic Wave Method .... 51
2.5.1.2 The Law of Resistance .... 55
2.5.1.3 Boundary Shear Stress .... 62
Kinematic Approach 62
Dynamic Approach 63
2.5.1.4 Entrainment Motion and
Critical Shear Stress .... 66
2.5.1.5 Flow Velocity 77
2.5.1.6 Turbulence 87
2.5.2 Overland Flow Erosion Studies 93
vi
9
such as a river. Here the soil will be carried until its final depo
sition in a reservoir, lake, delta, or ultimately, the ocean.
Usually a particle which is detached at the highest point of a
drainage area does not reach the river in the same rainfall event
since the erosion process usually is relatively a slow moving process
with respect to time. This is because the particle depends on an
external force (i.e., raindrop impacts, flowing water or wind) to be
detached and move downslope. If there is not any force capable of
moving the particle, it will remain in the same location. Since the
wind is excluded in this study, only during each rainfall event will
the particle have a downslope displacement. So the total distance
traveled by the soil particle will depend on the number of rain
storms, the specific rainfall characteristics and the soil character
istics.
The presence of rills and gullies in any area will depend on
the soil surface's properties, the steepness of the slope, and the
presence of vegetation. Therefore, one may find areas with highly
erosive soils in which gullies and maybe rills are absent due to the
lack of slope. On the other hand, it is possible to find gullies and
rills lacking even on very steep slopes if the soil is highly resis
tant to erosion (Ellison, 1947).
The raindrop impact effects in rills and gullies are usually
considered negligible compared to the flow discharge effects. This
is because the water depth in rills and gullies can be enough to sup
press the detachment capacity of the raindrop impact. In addition,
APPENDIX A
GENERAL NOTES IN THE EVALUATION OF THE ABSOLUTE
ERROR AND RELATIVE ERROR OF VARIABLES
The present development is based on Abramowitz and Stegun
(1972, p. 14). Let the dependent variable Yq be associated with
two independent variables and X2 by a general expression
Yd = f(X1# X2) = f(Xj) j = 1, 2 (A.1)
If Ydo, X10 and X20 are the corresponding approximations to the
true value of YD, Xj, and X3, then: the absolute error of those var
iables are
ayD = yD0 yD
AX1 = x10 X1
ax2 = X2q X^
and the corresponding relative errors are expressed as
AYp AX^ AX2
The absolute error of Yg can be related to the absolute error
of the independent variables Xj and X2 by the expression
AYd = JftX) AXi + JIM. AX2 (A.2)
u 9 1 3 X2
^ ^ ( X )
where ! represents the derivative of Yq equation with respect
to the variable Xj for j = 1, 2.
359
259
5.2.2 Estimated Relative Error of $
Following the same procedure and considerations used in the
evaluation of the relative error of qs and using the definition of
$ = 9S
_Y
% 3
 y)g d0
1/2
(3.16)
the estimated relative error of $ may be expressed as
A$ Aqs
3 Ade
2 d
e
(5.5)
5.2.3 Estimated Relative Error of V
The flow intensity function for horizontal or nearly horizontal
slopes, T, was defined as
r
(4.8)
Using the same procedure, the estimated relative error of y'
may be expressed as
A r Ade A^0
V d z
e o
(5.6)
5.2.4 Estimated Relative Error of the Slope Correction Factor of 4'*
2_
The slope correction factor of y' was defined by Equation 4.9
as:
Vrs
Slope Correction = = SC(e,) = cose sine cot^ (5.7)
Tcr
122
Later, Onstad et al. (1976) considered that detachment by
rainfall and runoff were not evenly distributed and expressed
Equation 2.62 as
R = aQ Rei + (1 ao)30 VR(qp)1/3 (2.63)
where "a0" is a coefficient between one and zero. Also, Foster et
al. (1977a, 1977b) used an equation similar to Equation 2.62 in SI
units.
Williams (1975) also modified the R factor, but in this case R
was considered as a function of the runoff energy only
R = 95(VRqp)*56 (2.64)
where Vp has units of acre"ft and qp has units of ft^/sec. Fur
thermore, the soil loss equation was adapted to study watershed by
modifying the Kf, LfS, Cm, and Pf factors based on their weighted
values according to the drainage area (Williams and Berndt, 1972).
This form allowed computation of the erosion from an entire watershed
in one solution of the modified LISLE, or MUSLE, the name usually
given to Williams' approach in the literature. The author indicated
that the MUSLE can be used to evaluate soil loss from an individual
storm, on a daily, monthly or annual basis. Further testing of MUSLE
(Williams and Berndt, 1977; Williams, 1982) indicated good agreement
using annual basis results, but for individual storm events the re
sults have shown a tendency to overpredict erosion from small storms
and underpredict the effects of large storms. Also, flat slope
334
probability of erosion is around 50%. However, the required preci
sion in the prediction of su for obtaining reasonable errors of $
is significantly increased as p approaches one because the error of $
can be about ten times greater than the error in su for probabili
ties of erosion on the order of 0.995 or higher.
The explanation of why the error in su becomes so important
as the probability approaches unity is clearly illustrated by observ
ing the error of the predicted value of $ in point XXIII (S0 = 30%
and I = 3.65 in./hr = 93 mm/hr). Its relative error of $ is 29.2%,
the second largest error in this study, and evaluations of the data
corresponding to that point did not show any reason for that error.
The probability of erosion at point XXIII is almost unity
(p = 0.9976 using Equation 4.41 to obtain su = 0.276, and
p = 0.9969 solving for p on Equation 3.17, which gives su = 0.283.
Under this condition, the very small variation in su, Asu~ 0.007,
produces a very small change in p, Ap 0.0007. However, when the
value of p is introduced in Equation 3.17, the results on the left
side of the equation give:
0.9976
= 415.67
1 p
Predicted
1 0.9976
P
0.9969
= 321.58
1 p
Required for
perfect agreement
1 0.9969
161
A* = "i Np(n_1) (lp)nA = A "z np(n_1)(lp) =
N n=l n=l
since
n=
1
n=l
np
(n1)
(1p)2
(3.8)
and where A was defined as the average saltation jump of the grain.
To apply this approach to overland flow with rainfall it is
necessary to recognize its limitations and present the required modi
fications. First, the overland flow depth is usually very shallow
and the potential height that a grain can reach due to a given ini
tial detaching force mc^y be restricted by the near water surface and
the surface tension effects. Secondly, even under steady state con
ditions, with constant rainfall intensity, the overland flow is a
spatially varied flow which will change the absolute probability of
erosion of a grain as the grains move downstream.
The effectiveness of the raindrop impacts to detach grains is
also a function of the thickness of the water layer over the grains
(Palmer, 1963, 1965). This thickness is going to change as the water
flows downstream. Generally, the greater the thickness of the water
layer is, the lower the amount of grain being detached by raindrop
impacts. If the overland flow by itself is not enough to sustain the
grains in motion, the net result will be shorter travel distances
than the ones expected for open channels. Nevertheless, the combined
effect of overland flow and raindrop impact can carry great amounts
of solids. It is considered that overland flow is the principal
227
remembered that rainfall affects many parameters at the same time.
When rainfall and no rainfall conditions are compared, it is, of
course, assumed that all other fundamental parameters are kept the
same. In this case the considered parameter was the saltation length
of the grains as it is influenced by the vertical momentum flux or
lack of it in the rainfall and no rainfall cases.
Now, the rainfall mass flux and the frequency of the raindrop
impacts per unit area per unit time can both be considered by the
rainfall intensity parameter. Rainfall intensity represents volumet
ric rainfall discharge per unit area per unit time. Multiplying vol
umetric rainfall discharge by the water density gives the rainfall
mass flux per unit bed area. At any given time during a rainfall
event a certain number of raindrops will impact a designated area
during a certain time. The volume of these waterdrops divided by the
area and time of collection gives the rainfall intensity during that
time. Therefore, using the rainfall intensity parameter alone the
number of rainfall variables to consider is reduced by one. However,
it should be mentioned that the use of rainfall intensity represents
the timemean uniform rainfall conditions on a given unit area. As
the considered area becomes smaller, the errors introduced in the
measurement of the rainfall intensity may become significant due to
the loss of uniformity of the areal distribution.
Some researchers (i.e., Laws, 1941; Laws and Parsons, 1943;
Gunn and Kinzer, 1949; Chow and Harbaugh, 1965, and others) have
indicated that the velocity of raindrops as they impact the water
173
equivalent to Inequality 3.19 has not been accomplished yet due to
the extremely involved relationship it represents. However, he indi
cated that under certain conditions one or more quantities might
vanish or at least become constant. Useful knowledge could thus be
obtained.
One of the usual assumptions made by researchers who have used
this equation is the consideration of uniform flow conditions along
the longitudinal slope. This assumes that the streamlines are paral
lel to the bed with no fluid acceleration and hydrostatic pressure
distribution prevails in sections normal to the streamlines. For
horizontal or nearly horizontal bed slopes the flow conditions show
that these assumptions are appropriately used. However, Ulrich
(1987) has indicated that the use of these assumptions required the
additional consideration that the buoyant force, B0, is no longer
vertical. Instead, since the buoyant force must be normal to the
pressure gradient, it becomes Bocos0 and must be normal to the bed
which, of course, is parallel to the water surface in uniform flow
(See Figure 3.5). Consequently, the correction factor due to longi
tudinal slope and uniform flow changes to
Y s tan 6
(ys y) tan 4
(3.22)
This indicates that incipient motion will occur at lower bed
shear stresses than the ones predicted by Equation 3.21. Equation
3.22 seems to be better justified from a physical point of view than
362
Evaluation of Equation 3.23 using Kilinc and Richardson's (1973) data
gave practically the same value as Equation 3.22 because dh/dx was
always very small (of the order of 10"^). Figure B.l shows the
values obtained from Equations 3.21 and 3.22 for Kilinc and Richard
son's silty sand material.
The use of Equation 3.22 instead of Equation 3.21 in the pre
sent model may not require more consideration than already given in
this work. However, the small saltation length represented by the
rolling particles over the rough bed surface when f(0) approaches
zero, i.e. with Equation 5.18 using C7 = ys/(ys y), may need fur
ther discussion. As mentioned in the presentation of Equation 5.18,
f(@) cannot be considered zero over a rough surface on steep slopes
because the rolling grain will not be always in contact with the
grains of the bed surface. A small value of f(0) e may therefore
better represent this condition.
For the data used in this study the points corresponding to 40%
bed slope require evaluation of f(0) = e because the assumption of
f(0 ) =0 does not allow use of the models general equation (i.e.
Equation 4.37). The possible magnitude of e was studied using the
following considerations.
Let
SCU = 1 Ys (B.2)
Ys Y tan*
and
e1 = small value of SCU
90
turbulence in water or air (e.g., Laufer, 1950; Raichlen, 1967;
Richardson and McQuivey, 1968; Blinco and Partheniades, 1971; and
others).
In addition, the data from Yoon (1970) and Kisisel (1971) may
partially support a constant value of au/u near the bottom for the
turbulent flows. Other researchers have also indicated this based on
available data and/or from analytical developments (Taylor, 1932;
Laufer, 1950; Christensen, 1965). Nevertheless, the magnitude of this
value may change from study to study because the measuring techniques
differ from study to study.
There are other physical considerations which can change the
o / value near the boundary. Lyles and Woodruff (1972) have indicat
ed that an increase of surface roughness results in increases of the
u/U value too. They have also indicated that not only the size of
the roughness elements affect the value but also their shape and ar
rangement. Results from Kisisel (1971), and Blinco and Partheniades
(1971) also show some of the effects of the roughness on au/u near the
boundary. Unfortunately, there is no established relationship between
au/ and the roughness characteristics near the boundary.
The relationship between u/v* and Y+ obtained by Yoon or
Kisisel (for no rainfall conditions) generally agree with the values
obtained by other researchers like Laufer (1950), Richardson and
McQuivey (1968), and Blinco and Partheniades (1971). However, Yoon's
results plotted in a curve almost parallel to Richardson and McQui
vey 's curve. He considered that this difference may be due to the
difference in flow conditions used in each test (i.e., Yoon's results
295
values and the rate of change of the water depth with respect to the
longitudinal distance is on the order of 10"^ or less. Consequently,
Equation 4.28 can be applied to that data. The possible use of this
equation for rainfall intensities on the order of 9 in./hr (229
mm/hr) or higher and no water layer on that area needs to be tested
before accepting its use. If it is found to be valid, the assumption
of a water layer over the bed surface all over the area may also have
to be modified in order to include the very high rainfall intensity
effects.
The elimination of the coefficient C3 is based on statistical
err
means greater than 600%) and the fact that the total contribu
tion of Equation 4.27 to the definition of C* was less than 2%. The
tested data has a very limited range of flow conditions but as the
discharge increases the effect of Equation 4.27 will be to decrease
its effect on C*. Therefore, the assumption that rainfall effects
decrease the saltation length may not be correct or at least has not
been completely defined by the approach used on Equation 4.27.
The most probable reason is the geometrical limitations placed
on the saltation length. For very shallow flows the saltation length
is limited by the physical boundaries of the water surface and the
bed surface with its roughness elements. Consequently, the saltation
of the grain will approach rolling (i.e., motion in almost full con
tact with the surface) instead of the free jump the grain may make in
deep water flows. In the rolling process, the particle is practically
152
America, and the American Society of Agricultural Engineers are
published every year with many topics in soil erosion or related to
it.
General sediment transport books which may help the reader to
understand the general concepts are Vanoni (1975), Raudkivi (1976),
Simons and Senturk (1976), Yalin (1977), Graf (1984), and Middleton
and Southard (1984) and others.
III
0.0799
3.87
3.482
VII
0.0787
3.81
3.134
XI
0.0749
3.64
2.664
XV
0.0722
3.50
2.243
XIX
0.0716
3.47
1.597
XXIII
0.0716
3.47
0.990
IV
0.0785
3.80
3.419
VIII
C.0787
3.81
3.134
XII
0.0741
3.59
2.628
XVI
0.0741
3.59
2.301
XX
0.0741
3.59
1.652
XXIV
0.0741
3.59
1.024
2.011
1.809
0.05235
1.050
0.869
0.30144
0.944
0.691
0.57830
0.840
0.538
1.2071
0.795
0.366
1.8340
0.710
0.202
2.9963
1.532
1.379
0.12011
0.950
0.782
0.47646
0.782
0.573
1.0433
0.713
0.457
2.1596
0.720
0.331
3.0391
0.635
0.181
5.2715
00
61
smooth turbulent flows, originally presented by Keulegan (1938), in
which Savat rearranged by using the DarcyWeisbach equation (Equation
2.28) and the flow Reynolds number definition to obtain the expres
sion
_L = 5.75 Â£og(Ref(Cf/8)1/2) + constant (2.39)
C ^
cf
Julien and Simons (1985) also suggested the use of the equation
originally proposed by Keulegan (1938), but they used Blasius'
equation for this kind of flow. Their definition for hydraulically
0 5
smooth flow was that the viscous sublayer, 6 = 11.6 v (P/Tq) was
greater than three times the size of the sediment particles, ds.
When the thickness of the viscous sublayer is small compared to
sediment size the flow is considered hvdraulically rough and the
logarithmic equation also given by Keulegan (1938) was considered to
apply. This equation was presented by Julien and Simons as
(BÂ£,1/2
cf
= C = a^ tog(a2
(2.40)
where aj and a2 are constants. However, they used approximated
power relationships such as Manning's equation to express the
friction factor.
Thornes (1980) also presented a similar expression to Equation
2.40 to obtain the friction factor which was originally used by Wol
man (1955).
Savat (1980) considered the resistance to flow in rough super
critical sheet flow which is present on steep slope flows. However,
25
Ma = 75.3 1?5*2
(2.12)
where MA is rainfall momentum per unit area and has units of
(kg m)/(s m^) and I has units of nm/hr.
Park et al. (1983) proposed the use of the expression
Mt = 64230 I1'09
(2.13)
where Mt is the rainfall momentum per unit area per unit time
(kg m/s)/(ha hr), I has units of mm/hr and Ctm is the tem
perature correction factor.
Finally, the relationship between the total number of drops
collected in a unit area per unit time, Ndr0p [drops/im2 s)],
and the rainfall intensity I (mm/hr),
drop 154 I05
(2.14)
presented by Park et al. (1983) may also help in the future to im
prove relationships for the soil erosion process.
All of these equations presented here have certain conditions
in order to be used correctly. The reader is referred to the origin
al studies for more information.
2.4 Splash Erosion
2.4.1 Waterdrop Splash
The study of the waterdrop impact and the consequent waterdrop
splash was improved with the introduction of the high speed cameras.
With this equipment, the different conditions and the time sequence
of this process were studied in detail. Ellison (1950) originally
presented sequences of photographs about the waterdrop splash on soil
206
in which the timemean critical bed shear stress obtained from Equa
tion 3.25 is used to account for those slope effects. Consequently,
y' is related to y' by the expression
0
(4.9)
H'q = 'V' (cos e sin ecoW)
where
H'g = flow intensity function for sloping beds.
With the new definition the lower integration limit in Equation
3.40 expressed by Equation 3.37 becomes
\ = ("o + l/su)(Eh )m l/su
(4.10)
where n0 was already defined to be equal to 3.09.
By this definition, the flow intensity function is related to
the sediment load transport of Equation 3.41. Table 4.3 shows the
values of and calculated from Kilinc and Richardson's data.
4.2.9 Sediment Transport Data in Diagrams
Einstein (1950), Chiu (1972), and others used diagrams of $
versus to present their result in dimensionless form. This study
will also use the same way of representation. Figures 4.2, 4.3, 4.4
and 4.5 show those diagrams for Kilinc and Richardson's data. For
these figures and for most of the figures presented in this study the
legend presented on Table 4.4 will be used to describe each data
point with constant rainfall intensity. This table also presents the
42
BÂ¡ = momentum correction factor for the raindrop's velocity
distribution
B_ = momentum correction factor for the lateral flow velocity
distribution
qi_ = lateral flow discharge per longitudinal unit length
g = gravitational acceleration
h* = overpressure head induced by the raindrop impacts over the
hydrostatic pressure head.
Sf = friction slope
The overland flow equations which are a special case of the
channel flow, can be readily obtained by considering the discharge
per unit length, q = Am/B. Other considerations are that the
lateral flow vanishes, B = 1, and the area becomes A = h(1) = h.
Hence, q = 0mh and the equations are expressed as follows:
 Conservation of mass for overland flow
8h + 9mh) = d f) cos 6 (2.18)
at ax
 Momentum equation for overland flow
9(hm) +
at
a(Bh^m )
ax
gh(S Sf) g ^ [h(*. h cos e + hj]
(2.19)
+ 8i I Vd si n(0 + fl)
which for nearly horizontal beds gives S = sin e= tan e= SQ
351
The proposed model may be adaptable to nonuniform slopes, such
as concave or convex beds or a combination thereof, if the local data
at each location of interest are used. However, the observations
made of the probability of erosion approaching unity as the bed slope
angle approaches the angle of repose may limit the use of the pro
posed model at very steep slopes (i.e., 40 or steeper for these data
and soil conditions).
To avoid any confusion with the results presented in this chap
ter, the use of Equation 3.22 (buoyant force normal to bed surface)
instead of Equation 3.21 (buoyant force considered in the vertical
direction only) throughout the development of the present model is
briefly presented in Appendix B. The results obtained using this
concept are also presented. The coefficients are evaluated using the
same procedure previously discussed and the predicted values show a
slight improvement. The maximum relative error of $ at any point was
found to be 29.6% instead of 33.1% predicted by the present model.
The standard deviation of the estimated error slightly improves to
0.1800 instead of 0.1873, and the correlation coefficient also
improves very slightly from 0.9955 to 0.9958. Consequently, as
indirectly proved by using DuBoys' bedload transport equation, the
use of Equation 3.21 instead of Equation 3.22 to correct for the
longitudinal slope effect in the critical shear stress will only
produce smaller errors is also proved with the present model. The
general observations made throughout Chapters IV and V seem to
continue to be valid too.
391
Piest, R. F., J. M. Bradford, and M. J. M. Romkens, "Mechanisms of
Erosion and Sediment Movement from Gullies," in Proc. of the
Sediment Yield Workshop, USDA Sedimentation Laboratory, Oxford,
Mississippi, Nov. 2830, 1972, U. S. Dept, of Agriculture,
Agricultural Research Service ARSS40, 162176, 1975a.
Piest, R. F., J. M. Bradford, and G. M. Wyatt, "Soil Erosion and
Sediment Transport from Gullies," J. of Hydraulics Division,
ASCE, lOl(Hyl), 6580, 1975b.
Podmore, T. H., and L. F. Huggins, "Surface Roughness Effects on
Overland Flow," Trans, of the American Society of Agricultural
Engineers, 23(6), 14341439, 1445, i960.
Poesen, J., "Rainwash Experiments on the Erodibility of Loose
Sediment," Earth Surface Processes and Landforms, 6, 285307,
1981.
Quansah, C., "The Effect of Soil Type Slope Rain Intensity and Their
Interactions on Splash Detachment and Transport," J. of Soil
Science, 32, 215224, 1981.
Raichlen, F., "Some Turbulence Measurements in Water," J. of the
Engineering Mechanics Division, ASCE, 93(EM2), 7397, 1967.
Raudkivi, A. J., Loose Boundary Hydraulics, Pergamon Press, Ltd.,
Oxford, England, Second Edition, 1976.
Richardson, E. V. and R. S. McQuivey, "Measurement of Turbulence in
Water," J. of the Hydraulics Division, ASCE, 94(Hy2), 411430,
1968.
Riezebos, H. T., and G. F. Epema, "Drop Shape and Erosivity: Part
II. Splash Detachment, Transport and Erosivity Indices," Earth
Surface Processes and Landforms, 10, 6974, 1985.
Robertson, A. F., A. K. Turner, F. R. Crow, and W. 0. Ree, "Runoff
from Impervious Surfaces Under Conditions of Simulated Rain
fall," Trans, of the American Society of Agricultural Engineers,
9, 343346, 351, 1966.
Rogers, J. S., L. C. Johnson, D. M. A. Jones, and B. A. Jones, Jr.,
"Sources of Error in Calculating the Kinetic Energy of
Rainfall," J. of Soil and Water Conservation, 22(4), 140142,
1967.
Rose, C. W., "Soil Detachment Caused by Rainfall," Soil Science, 89,
2835, 1960.
Rose, C. W., "Rainfall and Soil Structure," Soil Science, 91, 4954,
1961.
261
5.2.6 Discussion of the Error of the Longitudinal Slope
Correction Factor
The estimated error obtained from Equation 5.10 is based on
using Equation 3.21 as the longitudinal slope correction factor for
the critical shear stress at incipient motion. It was indicated in
Chapter III that based on the assumption of uniform flow and the
direction of the hydrostatic pressure distribution, the longitudinal
slope correction factor should be expressed by Equation 3.22. This
led to presentation of the more general case relationship in Equation
3.23. If Equation 3.22 is used, the estimated relative error of the
longitudinal slope may be expressed as
ASC (0, 4>) <() A _
SC(0,4>) (ys y) sin^<Â¡> cote sin<Â¡> cos ^
Ys (5.12)
Y
e(tano s cot)
Ys ~ Y A0
Y c 9
1  tane cot
Y$ Y
Table 5.3 shows the estimated error of the longitudinal slope
correction factor using the approaches represented by Equations 3.21
and 3.22. These values are based on
by Kilinc and Richardson (1973), and the estimated relative error of
0 and 0 shown in Table 5.1. Results from Equations 5.10 and 5.12
show that the estimated relative error may increase significantly as
the bed slope angle 0 increases (i.e., as the longitudinal slope
correction factor decreases).
314
of bed slope and rainfall intensity effects in order to physically
describe the saltation length correctly.
Equation 4.28 also indicates that, for constant bed slope and
water depth, the saltation length increases as the rainfall intensity
increases. This is completely opposite to the earlier development of
the C^expression and Equation 4.27. However, the paradox may have
a logical explanation. Under no rainfall conditions and shallow wa
ter depth, the transport capacity of the flow is very limited. The
flow discharge is small and the frequent obstructions to the flow,
due to the roughness elements, do not provide a good initial acceler
ation path for the grain.
When rainfall is present the raindrop impacts generate a high
level of turbulence in the fluid and increase the transport capacity
of the fluid. Raindrop impacts also provide an additional force cap
able of detaching grains and allowing them to travel greater dis
tances. The roughness elements still can reduce the potential salta
tion length, but the possibility of the grain to move between and/or
over the roughness elements is increased.
All this can be observed in Figure 4.5, in which the data ($
vs.y' ) are presented by curves with the same bed slope. On each
curve, an increase in rainfall intensity (i.e., more raindrop impacts
per unit time) increases the sediment transport intensity function,
$, by a significant amount compared with the change of the slope cor
rected flow intensity function, y'. This phenomenon becomes more
0
evident as the bed slope increases because the bed shear stress
215
force depends on the relative velocity between water and grain. This
means that the drag force will initially increase as the distance
from the bed increases, but later on the force will decrease as the
grain velocity approaches the water velocity.
The maximum grain elevation will be reached when the buoyant
weight becomes the predominant vertical force capable of making the
grains vertical velocity equal to zero. After reaching its maximum
elevation, the grain begins to fall again toward the bed surface, but
its longitudinal velocity component will carry the grain downstream
for a certain distance. That distance is controlled by the drag
force and the buoyant weight of the grain. Again in this segment of
the saltation length the traveled longitudinal distance must be
expected to be inversely proportional to the grain size. When the
flow conditions are assumed steady over the area under study, the
saltation length becomes only a function of the grainsize as it was
described by Equation 3.10.
The longitudinal bed slope effects were not considered by Chiu
because the slopes in rivers are usually less than 1%. For overland
flow, the bed slopes can very well be on the order of 40% or more.
Therefore, it is expected that the bed slope may affect the average
length of saltation, l, previously defined by Equation 3.10 as
l = Ci/de. As the slope increases the buoyant weight decomposes
in a component normal to the bed surface and in a component tangen
tial to the bed surface. This results in a reduction on the magni
tude of the acting force which shortens the jump length to Wcos9
O 10 20 30 40 50 60 70
tan0, %
Figure B.l. Correction Factor for Bed Shear Stress due to Longitudinal
Slope for Kilinc and Richardson's Silty Sand Material.
97
Walker et al. (1977) compared the magnitude of the erosion
caused by overland flow with rainfall (at different energy levels)
and without rainfall, but with the same total discharge at the down
stream end of the sloped surface in both cases. Their results showed
that raindrop impact was a powerful agent promoting soil transport
and preventing rill formation. They also showed that the bedload
movement was important in the transport of sand grains, even when the
runoff was disturbed by raindrops.
Then Walker et al. (1978) studied the effects of rainfall in
tensity, kinetic energy and bed slope on the characteristics of the
eroded material. The particles smaller than 31 pm were found to be
most readily mobilized and behaved as a suspended load. The parti
cles between 31 pm and 250 pm were transported slowly, much apparent
ly as saltation bedload while the 0.25 mm to 4 mm particles were
transported rapidly with the grains tending to move as rolling bed
load. Finally, the particles greater than 4 mm behaved as a lag
gravel layer (armouring) which became concentrated at the surface and
resisted erosion.
The results also showed that the effects of raindrop impacts
within the flow were more important in promoting transport of solids
than the aerial component due to the raindrop splash. They reported
that when overland flow had developed, the transport of particles was
directly related to rainfall intensity and the variations in the
rainfall energy that were associated with variations in raindrop
impact frequency. However, any increase in rainfall energy due to
17
techniques to measure the fall velocity they were able to work from
drop sizes so small (about 0.75 mm = 0.029 in.) that the Stokes Law
was obeyed to up to (and including) drops large enough to be mechan
ically unstable (about 6.1 mm = 0.24 in.). This work was done under
controlled conditions in stagnant air at 760 mm Hg pressure, a tem
perature of 20C (68F) and 50% relative humidity. The new observa
tions resulted in generally larger values than those found by other
researchers but approached more to the values obtained by Laws
(1941). The new values were measurably smaller than Laws' values.
The overall accuracy of the drop massterminal velocity measurements
of Gunn and Kinzer's study was better than 0.7%.
There are other studies dealing with the behavior of the fall
ing raindrop. For instance, Blanchard (1950) studied the growth of
larger waterdrops caused by collision with small drops, the breakdown
of larger waterdrops and the deformation of the waterdrop with time.
Jones (1959) considered the shape of the raindrops during rainstorm
events and concluded that there was basically a mean shape which
varied consistently with the mass of the raindrop. However, he also
observed that the shape was the result of oscillations about a mean
and that the tilt observed in the raindrop's major axis was associa
ted with the wind speed and its prevailing direction in the atmos
phere at the moment the measurements were taken. Likewise, Jameson
and Beard (1982) studied the oscillating forms of the freely falling
raindrops. Epema and Riezebos' (1984) study indicated that the
199
16% finer than 53 urn. These values show that the assumption of using
the grain size distribution of the original soil equation to the
grain size distribution would not significantly affect the prediction
for sediment transport due to overland flow with rainfall. It also
indicates that the use of the effective grain size, de, to repre
sent the grain size distribution of the transported sediment may be
justified.
4.2.2 Effective Grain Size Evaluation
Data from the sieve analysis of the original soil was used in
Christensen's (1969) expression for de evaluation
de=_!_= I (4.1)
^dF J:ns
Jo d j=l i3 dj_! tn(dj/dj_1)
where
d = grain size
F = fraction (by weight) of the sediment that is finer than d
ns = number of straight lines into which the grain size distri
bution curve is divided
j = arbitrary segment of the curve
The calculated value of de was 145 urn (4.76 x 10"^ ft)
which corresponds to an ordinate of about 31% finer. This de value
is very close to the d35 (grain size with 35% of the material
being finer than that size) proposed by Einstein (1950) to determine
flow intensity function, V. The de value is also in the middle of
the range d10 < de < d50 in which Christensen demonstrated that de
158
Einstein (1950) stated that the time to detach the grain from
the bed is proportional to the time required for a grain to fall a
distance equal to its own diameter at a velocity equivalent to the
steady settling velocity of the grain (vs). Following this
approach, t^ is expressed as
tl=A3(de/vs) (3'3)
where A^ is a dimensionless constant.
The value of vs may be found from the equilibrium condition
of the settling grain, which is a balance between the buoyant weight
and the drag force of the grain falling on stagnant water.
Weight = Drag
A2de3(Ys _Y) = CD B1 de2(YVs2/29) (3,4)
where
3
A2de = v0^ume 9rain with effective grain size dg
Cq = drag coefficient
2
Blde = area the ettective grain projected in direction of v$
ys = unit dry weight of soil grain material
Solving for the settling velocity of the grain, vs, and
substituting it into Equation 3.3, tj is expressed as
Yc 1/2
tl = A3 [Cd de/(g ('f 1))] (3.5)
where
A3 = A3 (B/2A2))ly/^2
A3 is a constant for the given soil particles represented on dg.
Generally, for very turbulent flows, is assumed constant.
112
The Y a1in equation was originally proposed as an equation to be
used for overland flow by Foster and Meyer (1972a). Since then it
has been used by other researchers (e.g., Khaleel et al., 1979; Beas
ley et al., 1980; Neibling and Foster, 1980; Foster et al., 1981;
Park et al., 1982; Foster, 1982; Dillaha and Beasley, 1983 and
others) to simulate sediment transport in small watersheds as part of
their computer models (e.g., CREAMS, ANSWERS).
Foster and Meyer (1972) selected this equation among several
published sediment transport equations because it was not too empiri
cal and seemed most applicable to conditions of concentrated shallow
flow associated with upland erosion. In addition, most of the as
sumptions made on the development of the equation still were valid
for overland flow. Only the hydraulic variables (hydraulic radius
and bed slope), soil parameters (grain size, specific gravity), and
critical shear stress from the Shields' diagram were needed to solve
this simple equation. They mentioned that, even though Yalin's equa
tion was developed for cohesionless particles, it appeared to be
applicable to transport of particles detached by rainfall and runoff
from cohesive soils given that these particles are detached in clus
ters or aggregates whose runoff transport behavior and size is simi
lar to that of sand.
Foster and Meyer modified the equation in order to consider
mixtures of particle sizes. This allowed the predicting of both the
total transport rate and also the particle size distribution of the
transported material. There are other modifications which have been
348
ha = de + 1.3dE = l + 1>3 E
de dg de
where
(5.20)
de = effective grainsize of bed material
dÂ£ = effective grainsize of silty sand material used in this
study = 145 urn = 4.76 x 10"^ ft
In this form the model may be better defined for any bed material.
This relationship needs corroboration from data obtained using dif
ferent effective grainsize materials. For the moment this relation
ship may be used for bed materials with effective grainsize similar
to the one used in this study without generating a significant error
of the predicted value.
The time mean bed shear stress is perhaps the most critical
parameter and to be used in this model. Its value directly affects
Vq and p value which will be used in the prediction of *. The value
of T*0 used in this study was obtained from the evaluation of the
steady state longitudinal momentum (dynamic) equation using all of
the terms as expressed by Kilinc and Richardson (1973). This is the
only form in which the rainfall effects can be really accounted for.
As presented in the literature review, many researchers have used
shorter versions of the momentum equation in order to evaluate .
This is not recommended for the study because the errors in the pre
diction of $ will be large and since is expected to include the
rainfall effects on the detachment and transportation of soil parti
cles through this very shallow water flow. The use of the total lon
gitudinal momentum equation to evaluate T0 brings the problem of
265
errors shown in Tables 5.1 and 5.2. Table 5.4 also shows the estima
ted error of the transport rate obtained from DuBoys formula and by
using Equation 3.21 instead of Equation 3.22 to obtain Tcrs. The
largest relative error of this transport rate is about 2.8% for the
data point I(S0 = 5.7% and I = 1.25 in./hr = 32 mm/hr) which has an
estimated relative error of qs, Aq$/qs, of 11.2%. All data points
show the Aqs/qsvalues higher than the transport rate relative error.
Consequently, the use of Equation 3.21 instead of Equation 3.22 in
estimating the transport rate observed by Kilinc and Richardson
(1973) may not generate a significant error in this study
The use of Equation 3.22 to obtain tcrs may create certain
inconsistencies with some of the data from Kilinc and Richardson
(1973). This equation indicates that ^crs = 0 for the 40% bed slope
data and, theoretically, a large degradation or motion of all grains
"in mass" should be observed for every rainfall intensity. The study
of the rate of change of the crosssectional average water depth with
respect to the longitudinal distance is of the order of 10^ or
less which leads to the consideration of the buoyant force being
nearly normal to the bed slope.
There are some possible reasons for not observing the expected
large degradation. One of them is that the soil has some cohesion
which provides an additional resistance to erosion. The shallow
overland flow conditions may have also increased the effect of hy
draulic roughness elements of intercepting the moving grains and
99
Meyer (1981) presented the simple expression to relate the soil
erosion (EÂ¡) on intern'll areas to the rainfall intensity (I) as
Ej = amIbm (2.57a)
where am and are empirical coefficients. He found that the
exponent "bm" decreased from slightly above 2.0 for soils with low
clay content to about 1.6 for soils with about 50% clay. That change
was approximated as
bm = 2.1 Clf (2.57b)
where Clf is the clay fraction percent.
The "am" coefficient was used to express the relative inter
rill erodibility of the different soils in their bare, tilled condi
tion or to reflect the different cropping conditions. He also found
that for highclay soils, the relative erodibility was also a func
tion of rainfall intensity.
Singer and Walker (1983) followed the works done by Walker et
al. (1978) and Moss et al. (1979) but considered the presence of a
straw cover over the soil. It was found that as the overland flow
increased the airsplash was reduced but the total soil loss in
creased. This was explained by the concept of rainflow transporta
tion already presented by Moss et al. and because of the changes in
the splash characteristics associated with changes of the water depth
(e.g., Palmer, 1963, 1965). Singer and Walker's results showed that
the rainflow transportation accounted for at least 64% of the soil
transport and the airsplash accounted for no more than 25% of the
soil transport.
286
Table 5.9 ANALYSIS OF VARIANCE
Source of
Variation
Degrees of
Freedom
Summation
of Squares
Mean
Squares
Ftest
Val ue
Regression
4
73.1345
18.4184
525.14
Residual
19
0.6664
0.03507
Total
23
73.8009
3.2087
Multiple coefficient of determination = 0.9910
Multiple correlation coefficient = 0.9955
Standard error of estimate = 0.1873
Mean value of in $ observed = 1.2449
Mean value of in $ predicted = 1.2405
Sample standard deviation of Â£n $ observed = 1.7913
Sample standard deviation of Â£n $ predicted = 1.7897
30
Harlow and Shannon (1967a, 1967b) solved the NavierStokes
equations for the waterdrop impacting a water layer phenomenon by
neglecting the viscosity. Also, surface tension effects at the water
surface were not considered. Solutions were presented for waterdrop
impacts onto a flat plate, into a shallow pool and into a deep pool.
The changes in splash configuration with respect to time were pre
sented for each case. Information about the pressures, velocities,
droplet rupture and effects on compressibility were also presented
for each water depth studied.
Wenzel and Wang (1970) used a different numerical approach than
the one used by Harlow and Shannon (1967a, 1967b) and included the
surface tension. Their results only consider the initial stages of
the waterdrop impact into stagnant water due to limitations in the
time of execution of the program and economic restrictions. These
initial stages included the period of time in which the waterdrop
impacts the water layer and the water moves radially outward. The
inward direction motion of the water was not included in the study.
Their results included a maximum impact pressure model and a quanti
tative discussion of pressure distribution, boundary shear, surface
tension effect, free surface configuration and various forms of ener
gy and their transformation during the impact process. Theoretical
results from the computer solution were successfully verified with
their experimental data of the impact pressure at the bottom surface
of a pan at various water layer depths. They used waterdrops of
various sizes falling at different impact velocities. Wenzel and
249
standard deviation of the estimated value of C3 was more than six
times the estimated value itself which means there is no statistical
basis for using the coefficient in the model. More discussions about
C3 will be presented in the next chapter.
After the C3 coefficient is eliminated from Equation 4.34,
the C+ function becomes
2.2 x 10'
ft"2 (4.36)
_ tane
in
tan
J
1 
C2 cos e
+ C4 y Bj Vt costil
de
(YsY)gde f(en)J
Considering all the available data and the aforementioned restric
tions, the computer program was used to find the value of the coeffi
cients. Equation 3.17 was solved for $ using the definition of C*
given by Equation 4.36. The values of Bj = 1, Q = 0, and f(en) =
0.54 were also used in the evaluation of C+. Following this reason
ing $ becomes the independent variable to be predicted by this model
using the equation
$ = (4.37)
1 p C* CD1/2 de2
The values of $ vary three orders of magnitude. Also the procedure
for solution of the coefficients is nonlinear. For convenience, the
natural logarithm of Equation 4.37 was therefore used in the computer
program.
11
that the raindrop impact was the driving force in transporting soil
in thin surface flows (sheet flows) to rills.
There are many considerations about the soil erosion process by
rainfall which have not been indicated in this section. It is better
to review them individually in order to understand this process from
single contributions of the factors and then joining them into a gen
eral soil erosion process description.
2.2 Initial Studies
Soil erosion has been studied extensively during the last half
century mostly due to its importance to agriculture and food produc
tion. Before the 1930s, the soil erosion problem was recognized but
not considered as a major problem. Therefore, there was not much
written about it and most of the literature available came from Euro
pean studies which did not apply directly to many of the conditions
found in the United States.
During the 1930s, there was an increased need for studies re
lated to soil erosion. It was realized that some of the most produc
tive lands were removed from agricultural production because the
water from rainfalls and the wind was carrying away the fertile top
soils and nutrients which the plants needed. Since there was not
much knowledge about the erosion process, the initial studies were
basically concentrated on collecting data which could help to estab
lish the magnitude of the problem and in studies to find some alter
natives or conservation practices to control erosion. In addition,
136
conditions. The soil detachment was determined from an equation
which considered the raindrop impact pressure estimates (Wang and
Wenzel, 1970) and reduction of soil detachment as the bed slope
increased due to decrease on the normal component of the raindrop
impact force. The raindrop impact velocity and size distribution,
and reduction of impact pressure as water depth increased were also
considered by this equation. The sediment transport capacity was
related to the stream power T0 0m) of overland flow. The net soil
loss rate predicted by the model corresponded to the smallest differ
ence between sediment transport capacity and soil detachment rate.
Recently, Sharma and Correia (1987) presented an equation for
soil erosion on forested hi 11 si opes (a mixed pine forest) on an indi
vidual storm basis. Starting from basic equations and after a few
simplifying assumptions, they arrived at the equation
qse = 0.58 Kfe x2 S2*1Vr (2.77)
with r = 0.933, where
qse = total soil mass per unit width (of the storm event)
Vr = volume of overland flow per unit surface area
Kfe = soil erodibility factor.
The application of the equation to a different environment requires
obtaining new coefficient values, to the ones reported as 0.58 for
the roughness coefficient and 2.1 for the bed slope exponent obtained
185
Now, let
B2 =
'2 .
1
1/2
(3.31)
Aj Eh (\/  t0  ) + cot
k ,
Substituting this into Inequality 3.30 and rearranging results into
(3.32)
(1 + n su)2 > B22 ^(1 + su2)
From this inequality the erosion criterion is obtained based on the
normalized velocity fluctuations and the average bed shear stress.
Erosion will take place when
1/2
and
^2
n = n > _
+ su
B2
n = n <
 su
(1 + s/)
(3.33)
crs
(1 +s/)
1/2
(3.34)
In open channels, the probability density of n is usually as
sumed to be distributed according to the Gaussian law. For overland
flow with rainfall there are some studies (e.g., Barfield, 1968;
Yoon, 1970; Kisisel, 1971, and Shahabian, 1977) which have measured
the velocity fluctuations in overland flow but, not as close to the
bed surface as is needed in this study. Indications of these studies
may support the assumption of using the Gaussian distribution to
describe the velocity fluctuations in overland flow with rainfall.
Therefore, it is assumed that the velocity fluctuations and n are
Gaussian distributed.
76
AlDurrah and Bradford (1982b) presented a description of the
soil detachment by raindrops impacting saturated soil surfaces based
on the soil shear strength as measured by the fallcone method al
reac(y mentioned in their previous publications. They also related
the angle at which the splashed particles are ejected from the
impacting area with the soil shear strength using an experimental
approximation (r^ = 0.93). Not surprisingly, low soil strength
resulted in a larger cavity at the soil surface, with a large accumu
lation of soil in a surrounding protuberance, a greater detachment of
soil particles due to the shear stress of the radial flow, and a
greater soil splash detachment angle with horizontal.
Nearing and Bradford (1985) continued the studies of AlDurrah
and Bradford. The new stud(y showed that with a correction in the
fallcone strength used in Equation 2.46, by reducing the fallcone
strength as a function of the triaxial friction angle, the resultant
relationship had the same slope coefficient value (a2) for all four
soils tested. They also indicated that the fallcone method alone
can predict the initial splash phase of cavity deformation upon im
pact, but it overpredicts the resistance of the soil to subsequential
lateral jetting of water. The results of the triaxial test alone, as
a measure of the strength and prefailure deformation properties of
soils, were also reported to be not good predictors of the soil
resistance to splash either.
Then Nearing et al. (1986) presented their results about the
waterdrop impact force and its changes with time. Four waterdrop
LIST OF SYMBOLS
A = cross sectional area of water flow
A0 = surface area exposed to falling raindrops
A} = constant of particle area
A2 = constant of particle volume
A3 = A3 (/(2A2) = constant
A3 = dimensionless constant
AÂ¡ = increment of surface area
a = 2.5
a0 = coefficient between 0 and 1 used by Onstad et al. (1976)
aj = constant
a2 = constant
a^ = thickness of the bedload transport layer, assume twice the
size of sediment particles
ad = kl = constant in DuBoys formula
af = coefficient relating detachment capacity to transport
capacity of flow
aÂ¡ = coefficient which depends in soil characteristics
a^ = coefficient used in discharge per unit width equation of the
kinematic wave method
am = empirical coefficient
ap = constant in velocity profile equation
ar = coefficient to relate rainfall intensity to the roughness
coefficient, K
ay = 2.45 x(y/ys)0,4
xv
120
vegetation, mulch, etc.) to the soil loss from the standard plot of
tilled and continuous fallow conditions with the same soil, slope and
rainfall. The factor accounts for the interrelated effects of cover
material, crop sequence, productivity level, growing season length,
cultural practices, residue management and rainfall distribution.
The evaluation fo the Cf factor is often difficult because of the
many existing cropping and managing systems and the variability of
the rainfall erosivity distribution in a given region or area.
Wischmeier (1960) presented an example of the evaluation of this fac
tor. Wischmeier (1975) also presented values to be used on undis
turbed areas, i.e., wooded, ranged or idle lands. The Cm values
are available in tables (Wischmeier and Smith, 1978) for general use
and the values are between 1.0 (bare soils) to 0.01 (some cover
material is protecting the soil surface).
The erosion control practice factor, Pf (dimensionless), is
the ratio of soil loss using specific practice compared with the soil
loss using upanddownslope farming (Smith and Wischmeier, 1957;
Wischmeier and Smith, 1965, 1978). The erosion control practices
usually included by this factor are contouring, contour strip crop
ping and terracing. Other control practices like conservation till
age, crop rotations, fertility treatments and retention of residues
are important but they are included in the cropping management factor
already described. The Pf values are presented in tables and show
some variation in their values with changes in bed slope (Wischmeier
and Smith, 1978).
338
describe the variation of the von Ka'rmcfn coefficient, k, in order to
substitute the original assumption of k = 0.40 (constant). Einstein
and AbdelAal (1972) proposed
parameter
(q Se vl1/2
where,
vs = settling velocity of grainsize d3$ (Einstein's
35 effective grainsize)
No corrections have been proposed for Chiu's model at very high
sediment rates. However, the region in which this very high proba
bility of erosion might occur in laboratory flumes and rivers is
where the flow intensity is high enough to create antidunes on the
bed surface, which means supercritical flow. For the overland flow
with rainfall considered in this study, the raindrop impacts in this
very shallow flow may not favor the anti dune formations. Neverthe
less, the turbulence generated by these raindrop impacts can provide
enough energy to detach and then help to transport very high sediment
rates equivalent to those rates in which the antidunes are formed.
Kilinc and Richardson did not present any indication of the presence
of bed forms on the bed surface or the degree of turbulence in their
overland flow study. Therefore, no further development in bed rough
ness, bed forms, and their effects on the su and p values can be
presented here.
159
Introducing Equation 3.5 into Equation 3.2 yields:
p[g( 1)]
A! A3 V/2
1/2
Ne =
(3.6)
3.2.2 Evaluation of Nd
The number of particles deposited per unit time and unit bed
area, Nd, is defined by
*. a2 V
where
(3.7)
qs = volume of particles with size de transported per unit
width of bed per unit time. qs = gs/Ys
gs = dry weight of sediment transported per unit width per
unit time
Â£* = average total distance traveled by a grain before it is
finally at rest U* may consist of several saltation
jumps)
3.2.3 Evaluation of Â£*
Einstein (1950) and Chiu (1972) evaluated Â£*, the average total
distance traveled by a grain, based on the absolute probability that
a particle is eroded. This absolute probability of erosion, p, was
also defined to be equal to the fraction of bed area where erosion
takes place at any time. Their analysis considered N particles pass
ing through the cross section aa (see Figure 3.1). Assuming all N
particles originate from the area & x 1 upstream of section aa, then
the average length traveled by one particle will be
23
Soil Loss Equation which has been widely used to predict soil loss
from laboratory and field areas during the last thirty years.
Elwell and Stocking (1973) used the expression originally
developed in Hudson's masters thesis (1965) for their Rhodesia,
Africa, region
KEA = 29.82 12^51
where KEA has units of (J/m2) and I has units of mm/hr.
Carter et al. (1974) proposed the expression
(2.7)
KEa = 429.2 + 534.0 I3Q 122.5 I302 + 7.8 I303 (2.8)
where KEA has units of (ft tons)/(acre in.) while I30 follows
Wischmeier and Smith's (1958) definition and has units of in./hr.
McGregor and Mutchler (1977) presented their expression as
KEa = 1035 + 822 exp(1.22 I30) 1564 exp(1.83 I30) (2.9)
where KEA and I3q have the same units as Wischmeier and Smith (1958).
Kneale (1982) obtained for small rainfall intensities
(0.1 mm/hr < I < 7 mm/hr) the expression
log KEt = 0.90 + 1.25 I (2.10)
where KEt has units of J/(m2 hr).
Park et al. (1983) have proposed the expression
KEt = 211070 Cte I1,16
(2.11)
305
Table 5.13 PREDICTED $VALUES AND THEIR ERRORS
Run
$
Predicted
Absol ute
Error
Relative
Error
(percent)
Standard
ized
Residual
I
0.00602
0.00176
22.58
1.367
V
0.02692
0.00310
13.01
0.653
IX
0.03439
0.01000
22.54
1.364
XIII
0.06390
0.01175
22.53
1.085
XVII
0.07999
0.00533
 7.13
0.368
XXI
0.09141
0.00042
0.46
0.025
II
0.02768
0.00336
13.84
0.692
VI
0.1626
0.04041
33.07
1.526
X
0.2210
0.02007
8.33
0.464
XIV
0.3443
0.1164
25.27
1.555
XVIII
0.6596
0.1626
19.77
1.177
XXII
1.1951
0.1343
12.66
0.637
III
0.05973
0.00737
14.08
0.703
VII
0.3193
0.01787
 5.93
0.307
XI
0.5792
0.00091
 0.16
0.008
XV
1.1910
0.01610
1.33
0.072
XIX
1.9277
0.09364
 5.11
0.266
XXIII
3.8724
0.87610
29.24
1.370
IV
0.1251
0.00501
 4.17
0.218
VIII
0.4571
0.01933
4.06
0.221
XII
1.3032
0.2599
24.91
1.118
XVI
1.9694
0.1901
8.80
0.492
XX
2.3481
0.6910
22.73
1.378
XXIV
4.9451
0.3264
6.19
0.341
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SOIL EROSION BY
OVERLAND FLOW WITH RAINFALL
By
Raul Emilio Zapata
December 1987
Chairman: Dr. Bent A. Christensen
Major Department: Civil Engineering
The objective of this study is to develop a soil erosion model
for overland flow with rainfall based on physical concepts and obser
vations. The basic sediment transport equation used by the proposed
model is based on the modified Einstein equation for total load
transport of noncohesive materials in open channels as presented by
Chiu in 1972. The proposed model was developed as generally as pos
sible in order to be valid for the case of deep water flows (i.e.,
rivers and open channels) as well as for very shallow flows (overland
sheet flow) with or without rainfall.
The proposed model provides a stochastic point of view of this
random process, usually modeled using deterministic approaches.
Rainfall effects on the erosion process are mostly represented in the
changes of the boundary shear stress and in the local velocity fluc
tuation at the top of the grains. The timemean bed shear stress is
obtained from the longitudinal momentum equation including additional
xxviii
218
energy to move the grain with the water. Since the maximum elevation
that the grain can reach has been reduced, the total saltation length
of one jump must also be less.
It should also be expected that the velocity profile is being
affected not only by the roughness of the bed surface but also by the
proximity of the water surface. The roughness size of a nonuniform
soil is usually greater than the effective grainsize dg. On a
bare soil surface, the roughness to effective grainsize ratio
k/de is expected to be around 10. This means that the roughness
and the water depth are of the same order of magnitude. Therefore,
the potential saltation length of any grain may also be interrupted
or reduced by roughness elements in front of it. This means that it
will be necessary to have an initial water depth (hj) over the
grains before the grain can move between or over the roughness ele
ments. Before that flow depth is reached the roughness elements, and
the capillary and viscous forces may prevent saltation or any other
type of motion of the grains.
The initial depth required to move grains is expected to be
equal to or greater than the depression storage elevation required to
initiate overland flow. This elevation value is expected to be be
tween the effective grainsize and roughness elevation of that bed
surface (e.g., de < hj < k). However, for a bed surface with known
grain size distribution, that initial depth may be considered con
stant.
282
Table 5.8 DATA POINT WITH POSSIBLE LARGE ERRORS OF $
Data point
(and Run Number)
Slope
(percent)
Rainfall
(in./hr)
intensity
(nin/hr)
V
10
1.25
(32)
I
10
2.25
(57)
XI
15
3.65
(93)
XIV
20
2.25
(57)
XV
30
3.65
(93)
XVIII
30
2.25
(57)
XXII
40
2.25
(57)
49
account for the increase in pressure due to the raindrop impacts and
the vertical momentum influx of the raindrops. This overpressure
term was presented by Chen and Chow (1968) as
P* = BÂ¡ p I VD cos0 cos (0 + fl) = p g h^ (2.21)
where
P* = overpressure due to raindrop impact
p = fluid density
They considered that this overpressure was uniformly distribu
ted over the cross section except at the free surface where P* be
comes zero in order to have atmospheric pressure at the free surface.
This approach is based on the assumption that the total head, T^,
over a vertical cross section is constant or
2
Th = y cose + + a = constant (2.22)
Y 2g
where
P = y(h y) cose + h^
a = energy correction factor
y = distance from the bed surface to a location in the water
This assumption created a discontinuity in pressure at an infinites
imal small distance, Ay, from the free surface.
Grace and Eagleson (1965) have considered that the overpressure
distribution was linearly distributed from zero at the free surface
to a maximum value of 2P+ at the bed surface. They presented expres
sions for the overpressure term based on vertical momentum equation
290
Test for su = 0.31 (constant) the Student's ttest gives
0.58 0.31
0.0356
= 7.87 > t
(19,0.975)
2.09
Therefore, the data from Kilinc and Richardson (1973) do not
support the assumption of su being constant. The tested case of
su = 0.16 corresponds to typical values for deep waterturbulent
flow conditions (Chiu, 1972) and the case of su = 0.31 corresponds
to extrapolations of the values presented by Yoon (1970) and Kisisel
(1971) for shallow water with rainfall. Equation 4.42, with the cor
responding values of C5 and Cg, appears to be good enough to
represent the required suvalues in the solution of Equation 3.19
and predict the erosion rates reported by Kilinc and Richardson
(1973). More discussions of su and the coefficients C5 and Cg
are presented in Section 5.5 of this chapter.
The C2 coefficient used in the evaluation of C* has a small
SEE
ratio of 13.3%. The null hypothesis was tested on this coeffi
cient to verify if the saltation length is not affected by the water
depth. The null hypothesis was rejected on the 5% level of signifi
cance. Consequently, the water depth is expected to affect the sal
tation length of the grain. The value of C2 = 2.3 0.3 indicates
that for horizontal slope with no rainfall conditions the initial
water depth required to initiate water motion in this soil material
due to surface tension and viscous forces as well as roughness ef
fects is about 2.3 times the effective grain size diameter, d^.
This water depth corresponds to the grain size of d54 (54% of mater
ial finer than this size) or 334 pm = 1.09 x 10"3 ft. The error of
317
in Equation 4.28 in order to indicate the variation of the saltation
length. To illustrate this phenomenon the results of four data
points with almost identical depth are presented in Table 5.15.
Although the water depth is basically the same, the increase in
rainfall intensity (i.e., increase of raindrop impacts per unit time)
corresponds to an increase of the saltation length depth function.
The effects of the bed slope are not explicitly included in these
results because they are implicitly included in the water depth val
ue. Remember that the discharge increases as the rainfall increases,
and to have the same water depth, the bed slope must increase in or
der to increase the velocity of the fluid. Therefore, the only way
to see the bed slope effects in Equation 4.28 is to follow the curve
of constant rainfall intensity as the bed slope increases in Figure
5.11.
The bed slope effects on the saltation length are not fully
accounted for by Equation 4.28. By following Chiu's physically jus
tified definition of the saltation length, Equation 3.10, the forces
which will be acting on the grain at the moment of detachment will be
affected by the bed slope. As discussed in Section 4.3.1, the salta
tion length depends on these initial forces (lift, drag, and buoyant
weight) in order to reach the potential saltation lenth. The forces
acting at the moment of detachment will be smaller than in the hori
zontal case considered by Chiu because the critical condition to
reach incipient motion decreases as the bed slope increases. Equa
tion 4.13 was obtained based on that physical approach.
50
and the order of magnitudes of each term. For the horizontal bed
with no infiltration and vertical falling raindrops the overpressure
term becomes
P+ = 0.5 p I VD (2.23)
Other researchers have used the overpressure term when the
momentum equation was presented in their studies (e.g., Kisisel,
1971; Kilinc and Richardson, 1973; and Shahabian, 1977). Kisisel and
Shahabian studies also included the rainfall turbulence effect in h*
and following Grace and Eagleson's approach of linear overpressure
distribution the h+ expression was presented as
h* = g [ \ BI 1 VD COS0 C0S (e + + v'2(h)^ (2.24)
where
2
v1 (h) = variance of vertical velocity fluctuations at the
free surface.
However, Shahabian's results show that the overpressure term induced
by the momentum influx of the raindrops seems to be a constant addi
tion to the hydrostatic pressure except at the free surface where
both the hydrostatic and overpressure terms are zero. This was based
on measurements at locations between 0.05 to 0.70 the water depth.
The magnitude of this overpressure with respect to other terms
in Equation 2.19 is sometimes small and the overpressure term is usu
ally neglected. The other reason to neglect this term is the collec
tion of data for the evaluation of h* in special values of Vq
and O.
B INFLUENCE OF CHANGE OF DIRECTION OF BUOYANT FORCE
ON PROPOSED MODEL 361
C CONVERSION FACTORS 371
REFERENCES 373
BIOGRAPHICAL SKETCH 400
ix
H I
e* *o Dio/Tco
K = von Krman constant
X = instantaneous lift per unit area
Xj = constant for saltation length
p = dynamic viscosity of water
v = kinematic viscosity of water
VA = kinematic viscosity of air
Â£ = water surface angle with respect to the horizontal
p = mass density of water
ogrr = standard deviation of the estimated error
1 = soil shear strength
T0 = instantaneous bed shear stress
T0 = timemean bed shear stress
tcr = timemean bed shear stress when p = 10"3
Tcrs = timemean bed shear stress for sloping beds when p
Tf = timemean shear stress due to form roughness
g = timemean shear stress due to grain roughness
s = timemean shear stress at the water surface
$ = sediment transport intensity function
<(> = angle of repose
xx vi
322
fe) =
1 C7 tan 9
tan 4>
m
(5.18)
Since the rolling motion represents a very small jump, the
value of C7 would be approximately 0.99999, which would give
f(0) a e when the bed slope angle is equal to the angle of repose,
e is assumed very small compared to unity. It is not possible to
determine C7 exactly, so it is better to say C7 = 1.0 and consi
der the rolling process as grain motion with full contact with the
bed surface at all times.
The bed slopes usually found in rivers are about 2% or less.
For these cases, the correction factor on the saltation length due to
the bed slope suggested by Equation 4.13 is between 0.97 and 1.0.
This means that making this correction factor equal to unity in the
cases of very small bed slopes will not change the predicted sediment
transport significantly. For greater bed slopes, the correction fac
tor proposed by Equation 4.13 should be used in order to prevent lar
ger errors in the solution.
The ratio between the saltation length defined by Chiu and the
definition used in the present model can be obtained by multiplying
the correction factor due to bed slope, Equation 4.13, by the water
depth function, Equation 4.28. It can also be obtained from the C*
constant value of 2.2 x 10? ft"^ obtained by Chiu, divided by the
C^ definition of the proposed model, Equation 4.40. The saltation
length ratio represents the change of the average saltation length
135
respect to time the sediment continuity equation to obtain their
sediment yield equation.
Rose et al. (1983b, 1983c) considered that rainfall detachment,
sediment deposition and entrainment of sediment could occur simulta
neously at different rates. The rainfall detachment was associated
with the rainfall intensity while the sediment deposition was associ
ated with the terminal settling velocity of the grains. The entrain
ment of sediment was related to the stream power concept and the
efficiency of the bedload transport mechanism. After some mathemati
cal procedures the sediment continuity equation reduced to an ordin
ary differential equation with an analytical solution. Integration
with respect to the time of the resulting equation led to the total
sediment yield for the storm event.
Rose et al. (1983c) used their method in an arid zone water
shed. They obtained good agreement with the measured sediment con
centration with respect to time at the downstream end of the water
shed if particular values of two physically defined parameters were
used. These are detachment of soil due to rainfall and the efficien
cy of sediment entrainment by overland flow. They found that the
first parameter had greatest effects on sediment concentration early
in any runoff event while the second dominated later concentrations.
This allowed the investigators to obtain an almost independent
assessment of both parameters.
Gilley et al. (1985a, 1985b) presented a model to describe soil
erosion on interrill areas for steady state flow and rainfall
247
Finally, Figures 4.8 and 4.9 show a strange behavior of some
points. This must be due to measurement errors or inaccuracies.
These and other reasons will be discussed in an error analysis sec
tion in Chapter V.
4.3.3 Procedure Used to Evaluate the Coefficients
The C* and su evaluations are based on the evaluation of
Equations 4.34 and 4.35, respectively. Both equations contain coef
ficients which have to be obtained from the experimental data. The
data were obtained from the literature. As mentioned before, the
only data which provided most the necessary information were due to
Kilinc and Richardson (1973). Their study provided 24 data points to
be used in the evaluation of four coefficients required to evaluate
su. Therefore, the solution presented here will only have 18 de
grees of freedom which may not be enough to justify the value of each
of the coefficients. Each of them will be discussed on the next
chapter.
The general procedure was to try to evaluate all the coeffi
cients at the same time by using nonlinear least squares analysis to
the highly nonlinear equation (Equation 3.17) and the respective
definitions of C* and su. The mathematical background of the gen
eral nonlinear least squares method were orginally presented by Went
worth (1965). The least squares adjustment is obtained following the
Lagrangian multipliers method. This method is also explained in sta
tistical books like Deming (1943, the original Wentworth source),
Brandt (1976), and others. The general computer program which used
that method was written by Whitman (1982).
263
The ASC(e, <}>) values of both approaches indicate that the ex
pected error of this longitudinal slope correction factor will be the
same for the same value of SC(6, ) using any of the two approaches.
This is because the estimated relative error of 0 and are assumed
constant as the bed slope angle increases. However, for a given bed
inclination the relative error estimated from Equation 5.12 is signi
ficantly higher than the one estimated by Equation 5.10. This is due
to the higher rate of change of the longitudinal slope correction
factor with respect to the rate of change of the bed slope angle,
i.e., the slope of the curve obtained from Equation 3.22 is steeper
than the slope of the curve obtained from Equation 3.21.
Assuming that the value obtained by Equation 3.22 is the more
realistic value, the relative and absolute error between approaches
appear to indicate for steep slopes (S0 > 15%) the use of Equation
3.21 as the longitudinal slope correction factor is not appropriate
for uniform flow conditions. However, the uniform flow condition on
steep slopes will also indicate high bed shear stresses which will
give very small values of Tcrs/T0_ratio and the effect of using
either Equation 3.21 or Equation 3.22 to define xcr$ becomes negligi
ble.
As indicated in Section 3.3.1, the estimates of sediment trans
port using the deterministic DuBoys formula may show a relative error
of about 5.3% for a relative error of xcrs of about 100%. Table 5.4
presents the estimated relative errors of qs, $ V, and using
data from Kilinc and Richardson (1973), and the estimated relative
166
the saltation length of a single grain's travel will be affected,
too. Figure 3.3 shows the longitudinal slope effects on the salta
tion length of the grain. Basically, the buoyant weight of the grain
may be resolved into two components with the result that the detach
ment force changes in magnitude and direction. Consequently, the
initiation of motion can occur at lower bed shear stresses (i.e., at
lower flow conditions or lower values of L and D) than the ones
required for the case of horizontal beds.
To describe how the saltation length is affected by these new
conditions, it is necessary to discuss other general aspects of the
model. Therefore, for the moment it is better to describe the aver
age saltation length as
i = function (acting forces on the grain, bed (3.11)
slope, water depth, rainfall
parameters)
or using Equation 3.10 with being a function and not a constant,
Cj = function (bed slope, water depth, rainfall (3.12)
parameters)
Recognizing the new meaning of Cj in Equation 3.10 and intro
ducing it into Equation 3.9 yields
Nd
qsdp)
C1 A2
(3.13)
The new meaning of does not affect the shape of the equa
tion that Chiu obtained for in open channels.
3.2.5 General Equilibrium Transport Equation
Substituting Equations 3.6 and 3.13 into Equation 3.1 to obtain
the expression for the equilibrium transport condition yields
142
purpose that it could be linked to hydrologic and chemical transport
models and not requiring calibration or collection of data to deter
mine the parameter values. The model considered soil erosion or
deposition from overland flow areas, concentrated flow areas (i.e.,
rills only), and small impoundments. The basic equations used in
this model were already presented by Foster and Meyer (1972a, 1972b,
1975) and Foster et al. (1977a).
For the case of overland flow, the flow depth was estimated by
the Manning's equation and the bed shear stress acting on the soil
t0 was estimated by
 h S (NMb/NMc)9/1 (2.79)
where
N^b = Manning's roughness coefficient for bare soil
NMc = Manning's roughness coefficient for rough, mulch, or
vegetative covered soil.
The rill erosion rate was expressed as
DF = Kfc 1.35 (?0 icr)1,05 (2.80)
where Kfc is a soil erodibility factor for channel erosion and
the critical bed shear stress, icr, is obtained from Shields' diagram
(Figure 2.3).
CREAMS also provided empirical relationships for the calcula
tion of particle size distribution and the aggregate size distribu
tion of the eroded material based on the respective primary fraction
of sand, silt and clay sizes of the soil.
179
Sediment transport can modify the pressure distribution by
changing an assumed constant density fluid motion into a stratified
density fluid motion or by generating bed forms (e.g., dunes or anti
dunes) which can change the parallel flow assumption of uniform
flows. On very steep slopes the tendency to have supercritical flows
may help in the formation of anti dunes or chutes and pools. Conse
quently, the uniform slope and uniform flow assumption may not be
appropriate to be used in those cases. The pressure distribution and
the buoyant force direction will depend on the location along the
longitudinal slope where they are measured.
The fact that the specific conditions of local bed slope and
water surface angles define the value of the correction factor due to
longitudinal slope may create a problem in the application of Equa
tion 3.23. Only in cases like the ones represented by Equations 3.21
and 3.22 or cases where the water surface does not change very rapid
ly the validity of Equation 3.23 may be accomplished. An independent
study will be needed in order to verify Equation 3.23.
However, uniform flow on steep slopes usually produces a bed
shear stress which may be significantly greater than the bed shear
stress required to reach incipient motion of the particles carried by
overland flow or regular water courses. So, even though there is a
significant difference between the predicted correction values ob
tained from Equations 3.21 and 3.22, the net influence on the sedi
ment transport rate can be expected to be small.
281
rate of the soil does not significantly change as the bed slope in
creases. Therefore, the flow discharge is basically constant but the
water depth decreases. As mentioned before, the decrease of water
depth increases the sediment transport dependence on raindrop impact,
consequently, it may increase the fluctuations of the collected data.
Therefore, the amount of sediment being transported out of the area
may show less fluctuations as the bed slope decreases.
From the data used here it appears that rainfall influences the
relative error stronger than bed slope does. From the preceding con
siderations and the values presented in Table 5.2 and Table 5.4, it
appears that the points which might have larger $errors are the ones
presented in Table 5.8.
5.2.8 Other Possible Errors
Another parameter which merits further discussion is the water
depth of the overland flow. Kilinc and Richardson did not directly
measure this depth directly. They obtained a mean water depth value
from measurements of the water discharge and the spatial mean velo
city. This method may have provided values good enough for most of
their hydraulic considerations. The approach used here, however,
requires local water depth values rather than mean values. Under
sheet flow conditions, usually found in flow with flat slopes, the
mean water depth will suffice in most models. However, for the bed
slope conditions used by Kilinc and Richardson, water flow can gener
ate rills which will change the hydraulic roughness characteristics
of the area.
387
Meyer, L. D., and W. C. Harmon, "Susceptibility of Agricultural Soils
to Interrill Erosion," J. of Soil Science Society of America,
48, 11521158, 1984.
Meyer, L. D., C. B. Johnson, and G. R. Foster, "Stone and Woodchip
Mulches for Erosion Control on Construction Sites," J. of Soil
and Mater Conservation, 27(6), 264269, 1972.
Meyer, L. D. and E. J. Monke, "Mechanics of Soil Erosion by Rainfall
and Overland Flow," Trans, of the American Society of Agricul
tural Engineers, 8(4), 572577, 580, 1965.
Meyer, L. D. and W. H. Wischmeier, "Mathematical Simulation of the
Process of Soil Erosion by Water," Trans, of the American
Society of Agricultural Engineers, 12(6), 754758, 762, 1969.
Middleton, G. V., and J. B. Southard, Mechanics of Sediment Movement,
2nd Edition, Lecture Notes for Short Course No. 3 sponsored by
the Eastern Section of the SEPM, Providence, Rhode Island, March
1314, 1984, Society of Economic Paleontologists and Mineralo
gists, 1984.
Mihara, Y., "Raindrop and Soil Erosion," Bull, of the National
Institute of Agricultural Sciences, Series A, No. 1, 251,
T9FH
Mitchell, J. K., and G. D. Bubenzer, "Soil Loss Estimation," in Soil
Erosion, M. J. Kirkby and R. P. C. Morgan (Eds.), John Wiley
and Sons, Ltd., Chichester, Great Britain, 1980.
Moldenhauer, W. C., and J. Koswara, "Effect of Initial Clod Size on
Characteristics of Splash and Wash Erosion," Proc. of Soil
Science Society of America, 32, 875879, 1968.
Monke, E. J., H. J. Marelli, L. D. Meyer, and J. F. DeJong, "Runoff,
Erosion and Nutrient Movement from Interrill Areas," Trans, of
the American Society of Agricultural Engineers, 20(1), 5861,
T5TT.
Morgali, J. R., "Laminar and Turbulent Overland Flow Hydrographs," J.
of the Hydraulics Division, ASCE, 96(Hy2), 441461, 1970.
Morgali, J. R., and R. Y. Lindsey, "Computed Analysis of Overland
Flow," J. of the Hydraulics Division, ASCE, 91(Hy3), 81100,
1965.
Morgan, R. P. C., "Field Studies of Rainsplash Erosion," Earth
Surface Processes and Landforms, 3, 295299, 1978.
Morgan, R. P. C., Soil Erosion, Longman, Inc., New York, 1979.
j
Table 2.2 RELATIVE MAGNITUDES OF THE TERMS S, S^ S2, AND S3 IN TERMS OF Sf
Reference
so
Relative Magnitude (percent)
S
sf
S1
sf
s2
Sf
S;
3
F
Min
Max
Min
Max
Min
Max
Min
Max
Yoon
0.005
100.2
122.2
0.467
12.68
0.445
15.72
0.0112
1.157
Yoon
0.01
99.0
110.8
3.45
3.58
0.328
13.29
0.0116
1.193
Robertson et al.*
0.05
100**
0.144
0.27
1.08**
0.612
1.82
Robertson et al. (1966) results are based only on two sets of absolute value reported in their article.
They used Sf = S = sinwhile Yoon (1970) obtained Sf using Equation 2.20b.
**0nly one value was reported.
4=.
00
58
Another equation which is frequently used for the turbulent
flow regime is Manning's equation in English units (assuming R' = h)
0 = Li9 s h2/3
m Nm f
(2.35)
Robertson et al. introduced the DarcyWeisbach equation and solved
for Cf to obtain
8g nm2
1.49 h1/6
(2.36)
or
Sfl/10 ... V/5. c
( VRef)!/5 i.499/5 Ref1/5 Ref1/5
(2.37)
In this form the equation has a similar form to Blasius equation
(Equation 2.34). Robertson et al. (1966) reported that in one of
their three rough surfaces studied the value of the flow Reynolds
number exponent was equal to 0.20 = 1/5 with correlation coefficient
of 0.74 for the Ref range between 550 and 4500.
For larger Ref values the effects of viscosity and rainfall
are diminished and the friction factor is usually considered constant
for that bed surface. Consequently, the Chezy's equation can be used
since Cf no longer depends on Ref.
Yen et al. (1972) developed equations for the friction slope
(Sf), the total head slope (Sh) and the dissipated energy gradi
ent (Se) from the general equations presented in Yen and Wenzel
162
transporting agent of soil erosion and that raindrop impacts provide
the necessary turbulence or agitation of the water to keep the grain
moving by small saltation jumps.
The Einstein approach can be applied to overland flow if the
following two considerations are studied. First, the rainfall inten
sity is constant over the total area of study. This consideration
was presented before as one of the basic assumptions but the reader
is reminded of it here again. Secondly, the saltation length is
affected by the water depth because the raindrop impact detachment
decreases as the water depth increases and the proximity of the water
surface to the soil surface can reduce the potential saltation length
of the grain. So the changes in the saltation length along the lon
gitudinal slope (Ai/Ax) will be expected to be small if the rate of
change of the water depth along the longitudinal slope Ah/Ax is small
too.
Data from Yoon (1970) reveals that the rate of water depth to
longitudinal distance is of the order of 104 or less after the
initial first few feet of the longitudinal sloping bed. This means
that the rate of change of the saltation length with respect to the
longitudinal distance will be very small. Therefore, with the excep
tion of the section where overland flow begins (i.e., where the rate
of change of water depth to longitudinal distance can be signifi
cant), the saltation length can be taken as constant for certain
depth increments as it is shown in Figure 3.2.
If the depth is assumed constant for the longitudinal increment
under study, the probability of erosion will not be significantly
5.11 REQUIRED MINIMUM WATER DEPTH TO USE THE PROPOSED MODEL . 293
5.12 DATA POINTS WITH PREDICTED ERRORS LARGER THAN THE
ESTIMATED ERROR OF DATA 300
5.13 PREDICTED $VALUES AND THEIR ERRORS 305
5.14 SUMMARY OF DATA POINTS WITH LARGEST ERROR ON THE
PREDICTED MODEL SOLUTION 306
5.15 RAINFALL INTENSITY EFFECTS IN f(h/de) FUNCTION FOR
GIVEN WATER DEPTH 318
5.16 PREDICTED su AND pVALUES 333
B.l ANALYSIS OF VARIANCE USING IMPROVED METHOD (EQUATION
3.22) 365
B.2 STATISTICAL ANALYSIS OF THE COEFFICIENTS IN THE MODEL . 366
B.3 PREDICTED $VALUES AND THEIR ERRORS 367
B.4 PREDICTED su AND pVALUES OF THE IMPROVED METHOD 368
xi
148
25 mph (40 km/hr) detached about 2.7 times more soil than when there
was no wind.
Throughout this chapter the effect of vegetation, mulch or any
other cover on the soil erosion was presented. Their general effect
is that they intercept the raindrop before it reaches the soil sur
face and absorbs much of the raindrop's kinetic energy. After this
impact the raindrop may not be able to regain high velocities at the
time it reaches the soil. Consequently, the surface crust may not be
formed as fast as for the bare soil condition or may be present in
smaller surface areas. It will, therefore, allow high infiltration
rates and keep the runoff at a minimum. This leads, of course, to a
reduction of the soil erosion, even on steep slopes.
There are many more studies considering the effect of vegeta
tion or manmade covers on the soil erosion (e.g., Kramer and Meyer,
1969; Meyer et al., 1972; Lattanzi et al., 1974; Stocking and El well,
1976, and others). Each of them have demonstrated the effectiveness
of cover materials used under the various conditions studied by the
researchers (i.e., bed slope, soil type, rainfall intensity, and
others). The general results were similar to the ones already pre
sented throughout this chapter. The reader may be referred to the
original papers if more details are required.
2.8.2 Slope Gradient Effects
Early studies like Neal (1938), Zingg (1940) and others (e.g.,
Section 2.2) recognized the effect of the bed slope on the soil ero
sion process. Neal reported that the soil losses from a saturated
Fully developed
turbulent
velocity profile
Sym
Description
r,.g/ems
o
>
Amber
Lignite
Granite
Borite
Sond (Co
Sond (Kr
Sond (U*
Sand (Gi
(Shields)
sey )
omerl
.WES)
bert)
1 06
1 27
2 7
4 23
2 65
2 65
2 65
2 65
Turbulent
boundor y
loyer
Sond (Vononi)
Clo beod (Vononi)
Sond I White)
Sond in on ( White)
Steel hot (white)
Curve
Figure 2.3. Shields' Diagram for Incipient Motion Including Mantz Extended
for Fine Cohesionless Grains (based on Graf, 1984 and Mantz, 1977).
O
374
Barnett, A. P., and J. S. Rogers, "Soil Physical Properties Related
to Runoff and Erosion from Artificial Rainfall," Trans, of the
American Society of Agricultural Engineers, 9(1), 123125, 128,
1955:
Barnett, A. P., J. S. Rogers, J. H. Holladay, and A. E. Dooley, "Soil
Erodibility Factors for Selected Soils in Georgia and South
Carolina," Trans, of the American Society of Agricultural
Engineers, 8, 393395, 1965.
Beard, K. V., "Terminal Velocity and Shape of Cloud and Precipitation
Drops Aloft," J. of the Atmospheric Sciences, 33, 851864,
1976.
Beasley, D. B., L. F. Huggins, and E. J. Monke, "ANSWERS: A Model
for Watershed Planning," Trans, of the American Society of
Agricultural Engineers, 23(4), 938944, 1980.
Bennett, J. P., "Concepts of Mathematical Modeling of Sediment
Yield," Water Resources Research, 10(3), 485492, 1974.
Bisal, F., "Calibration of Splash Cup for Soil Erosion Studies,"
Agricultural Engineering, 31, 621622, 1950.
Bisal, F., "The Effect of Raindrop Size and Impact Velocity on
SandSplash," Canadian J. of Soil Science, 40, 242245, 1960.
Blanchard, D. C., "The Behavior of Water Drops at Terminal Velocity
in Air," Trans, of the American Geophysical Union, EOS, 31(6),
836842, 1950.
Blinco, P. H., and E. Partheniades, "Turbulence Characteristics in
Free Surface Flows Over Smooth and Rough Boundaries," J. of
Hydraulic Research, 9(1), 4371, 1971.
Blong, R. J., 0. P. Graham, and J. A. Veness, "The Role of Sidewall
Processes in Gully Development; Some N. S. W. Examples," Earth
Surface Processes and Landforms, 7, 381385, 1982.
Bogardi, J., Sediment Transport in Alluvial Streams, Akademiai Kiado,
Budapest, Hungary, 1974.
Bouyoucos, G. J., "The Clay Ratio as a Criterion of Susceptibility of
Soils to Erosion," J. of the American Society of Agronomy, 27,
738741, 1935.
Bradford, J. M., P. A. Remley, J. E. Ferris, and J. B. Santini,
"Effect of Soil Surface Sealing on Splash from a Single
Waterdrop," J. of Soil Science Society of America, 50, 1547
1552, 1986.
73
downstream end of the shallow water channel which represents the
downstream condition just before the overland flow reaches a main
stream because his flow conditions were also in the turbulent
regime.
Other researchers have developed expressions for soil detach
ment due to raindrop impact on unprotected bare soils, using differ
ent kinds of failure criteria (e.g., Ghadiri and Payne, 1977, 1981;
Cruse and Larson, 1977; A1 Durrah and Bradford, 1981, 1982a, 1982b;
Nearing and Bradford, 1985).
The study of Ghadiri and Payne (1977) did not produce satisfac
tory results in the breakdown of paper tissue and soil crumbs (dry
pieces of soil aggregate) by the stress produced by raindrop impact,
but it opened the doors for more promising research studies. Their
calculated critical shear stresses did not agree closely with values
found by other methods and varied systematically with the raindrop
size. Their discrepancies were related to the duration of the appli
cation of the shear stress and to the local shear stress concentra
tion at the periphery of the drop impact.
Then, Ghadiri and Payne (1981) studied the raindrop impact
stress by measuring the penetration of waterdrops through nylon mesh
es, by analysis of cinephotographs of splash and by the use of a
force transducer. Their results indicated maximum stress of 2 MPa
(290 lb/in.2) to 6 MPa (870 lb/in.2) acting for about 50 ps on the
perimeter of a circle. They also indicated that the effect of a 2 mm
(0.0787 in.) water film covering the nylon mesh seemed to reduce the
118
the soils in the United States. For soils where Kfvalues are not
available, a soil erodibility nomograph was developed by Wischmeier
et al. (1971) using five soil parameters: percent silt (2 urn to
50 urn) plus very fine sand (50 urn to lOOum), percent sand (0.1 mm to
2.0 mm), organic matter content, soil structure, and soil permeabil
ity. Recently, Mutchler and Carter (1983) showed that the Kf fac
tor has a periodic variation during the year which ranged from a high
of 169% of the average annual values to a low of 31%.
Topographic effects on soil erosion has been considered by the
USLE in two separate factors, Lf for the slope length effects and S
for the slope gradient effects. However, these two factors are often
evaluated as a single topographic factor LfS.
The slope length factor, Lf (dimensionless), represents the
distance from the point of origin of overland flow to the downslope
point where the slope decreases sufficiently for allowing deposition
to occur, or to the point where runoff enters a natural or manmade
channel, with respect to the standard plot length. Wischmeier and
Smith (1978) have recommended the following values for the M_
exponent
ML = 0.5
Ml = 0.4
ML = 0.3
ML = 0.2
for
So
> 5%
for
5%
V
CO
o
V
3%
for
3%
V
CO
o
V
1%
for
So
< 1%
Mutchler and Greer (1980) have recommended M_ = 0.15 for
slopes less than 0.5%, specially for saturated soil surfaces. Their
67
saltation and the smaller size particles will predominantly move by
suspension.
Usually particles transported in suspension are referred to as
the suspended load. The particles which move by rolling or saltation
are referred to in the literature as bedload. There is also the so
called washload which is made up of grain sizes finer than the bulk
of the bed particles and thus is rarely found in the bed of the
stream. These particles are usually washed through the section of
the stream. Some authors have considered washload and suspended load
as the same load in their sediment transport definition. The total
sediment transport load is referred to as the sum of bedload and
suspended load and/or washload depending on the definition used by
the authors. In this study, the total load will be considered as the
summation of bedload and suspended load with washload being included
into the suspended load.
The critical force required to begin the motion of particles
has been associated basically with two theories. First, Hjulstrorn
(1935; Graf, 1984) presented an erosiondeposition criteria based on
the crosssectional mean flow velocity (0m) required to move parti
cles of certain size. Figure 2.2 shows this basic erosiondeposition
criterion for uniform particles. It shows the limiting zone at which
incipient motion starts and the line of demarcation between the sedi
ment transport and sedimentation. The diagram also indicates that
loose fine sand is the easiest to erode and that the greater resis
tance to erosion in the smaller particle range must depend on the
cohesion forces.
394
Smith, T. R., and F. P. Bretherton, "Stability and the Conservation
of Mass in Drainage Basin Evolution," Water Resources Research,
8(6), 15061529, 1972.
Spilhaus, A. F., "Raindrop Size, Shape and Falling Speed," J. of
Meteorology, 5, 108110, 1948.
Stevens, M. A., and D. B. Simons, "Stability Analysis for Coarse
Granular Material on Slopes," River Mechanics, II, H. W. Shen
(Ed.), Water Resources Publications, Littleton, Colorado, 1971.
Stevens, M. A., D. B. Simons, and G. L. Lewis, "Safety Factors for
Riprap Protection," J. of the Hydraulics Division, ASCE,
102(Hy5), 637655, 197F:
Stocking, M., and H. Elwell, "Vegetation and Erosion: A Review,"
Scottish Geographical Magazine, 92(1), 416, 1976.
Tackett, J. L., and R. W. Pearson, "Some Characteristics of Soil
Crusts Formed by Simulated Rainfall," Soil Science, 99, 407413,
1965.
Taylor, G. I., "Note on the Distribution of Turbulent Velocities in a
Fluid Near a Solid Wall," Proc. at Royal Society of London,
Series A, 135, 678684, 1937:
Thornes, J. B., "Erosional Processes of Running Water and Their
Spatial and Temporal Controls: A Theoretical Viewpoint," Soil
Erosion, M. J. Kirkby and R. P. C. Morgan (Eds.), John Wiley
and Sons, Chichester, Great Britain, 1980.
Threadgill, E. D., and R. E. Hermanson, "Effect of Rainfall on the
Velocity Distribution and Tractive Force in a Triangular Open
Channel," Trans, of the American Society of Agricultural
Engineers, 12, 352355, 1969.
Todten, H., "A Mathematical Model to Describe Surface Erosion Caused
by Overland Flow," Z. fur Geomorphologie N. F., Suppl. Bd. 25,
89105, 1976.
Toy, T. J. (Ed.), Erosion: Research Techniques, Erodibility and
Sediment Delivery, Geo Abstracts Ltd., Norwich, England, 1977.
Ulrich, T., "Stability of Rock Protection on Slopes," J. of
Hydraulic Engineering, ASCE, 113(7), 879891, 1987.
USAWES, "The Unified Soil Classification System," Waterways Experi
mental Station Technical Memo No. 3357, U. S. Arrny Engineers
Waterways Experimental Station, Corps of Engineers, Vicksburg,
Mississippi, 1967.
392
Rose, C. W., J. Y. Parlange, G. C. Sander, S. Y. Campbell, and D. A.
Barry, "Kinematic Flow Approximation to Runoff on a Plane: An
Approximate Analytic Solution," J. of Hydrology, 62, 363369,
1983a.
Rose, C. W., J. R. Williams, G. C. Sander, and D. A. Barry, "A
Mathematical Model of Soil Erosion and Deposition Processes: I.
Theory for a Plane Land Element," J. of Soil Science Society of
America, 47, 991995, 1983b.
Rose, C. W., J. R. Williams, G. C. Sander, and D. A. Barry, "A
Mathematical Model of Soil Erosion and Deposition Processes:
II. Application to Data from an AridZone Catchment," J. of
Soil Science Society of America, 47, 9961000, 1983c.
Ross, B. B., V. U. Shanholtz, and D. N. Contractor, "A Spatially
Responsive Hydrologic Model to Predict Erosion and Sediment
Transport," Water Resources Bulletin, AWRA, 16(3), 538545,
1980.
Rouse, H., Elementary Mechanics of Fluids, Dover Publications, Inc.,
New York, 1946.
Rowlinson, D. L., and G. L. Martin, "Rational Model Describing Slope
Erosion," J. of the Irrigation and Drainage Division, ASCE,
97(IR1), 3950, 1971.
Savat, J., "The Hydraulics of Sheet Flow on a Smooth Surface and the
Effect of Simulated Rainfall," Earth Surface Processes and
Landforms, 2, 125140, 1977.
Savat, J., "Resistance to Flow in Rough Supercritical Sheet Flow,"
Earth Surfaces Processes and Landforms, 5, 103122, 1980.
Savat, J., "Work Done by Splash: Laboratory Experiments," Earth
Surface Processes and Landforms, 6, 275283, 1981.
Schoklitsch, A., Handbuch des Wasserbaues, Springer, Vienna (2nd
Edition, 1950, and English Translation by S. Shulits in 1937),
1930.
Schroeder, S. A., "Slope Gradient Effect on Erosion of Reshaped
Spoil," J. of Soil Science Society of America, 51, 401409,
1987.
Shahabian, H. L., "Turbulence Characteristics of Laboratory Generated
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Purdue University, West Lafayette, Indiana, 1977.
109
sitespecific conditions for which each relationship was derived.
Finally, Julien and Simons mentioned that the value of this exponent
increased when the formation of rills was observed on the soil sur
face.
Alonso et al. (1981) pointed out the obvious limitations of the
classic sediment transport equations recognizing that each equation
was derived for a limited range of sediment and flow characteristics.
The nine equations selected by these researchers were chosen for
their particular applicability to hydrologic modeling of agricultural
watersheds. The equations were evaluated against laboratory and
field data but none of them represented the entire spectrum of sedi
ment and flow characteristics.
The results indicated that the Yang (1973) equation was the
best for estimating the carrying capacity in the range of fine to
coarse sands. The Laursen (1958) equation provided a reasonable pre
diction of the carrying capacity of small water courses in very fine
sands and silts; however, its use was recommended with some reserva
tions. For overland flow, the Yalin (1963) equation may be used to
compute the sediment transport capacity. This equation showed satis
factory results for the range of particle sizes and densities which
are characteristic in most field situations. It also showed that it
could be used with confidence to predict transport rates of light
materials in water courses.
Alonso et al. also reported that although none of the three
selected equations performed well when the specific experiments were
307
5.3.4 Justification of the Least Squares Approximation Method
The use of a general mathematical procedure or method to solve
a problem does not insure that the solution is correct. It is neces
sary to verify that the methodology was used correctly and that all
of the assumptions of the method are satisfied. This study used the
least squares approximation method for the solution of nonlinear
regression analysis.
Ott (1984) indicates that there are three basic steps to follow
in order to use the general least squares method correctly. These
are
(1) Identification of the possible independent variables
(2) Consideration of the form of the (multiple) regression
model
(3) Satisfaction of the residual analysis of the model which
assumes
(a) The expected value of the estimated error must approach
zero
(b) The variance of the error must be constant
(c) The errors are normally distributed
(d) The errors are independent of the variables.
The first two steps deal mostly with the design of the model
configuration and the important variables which are necessary to
achieve good predicting values. Those two steps do not need further
discussion. However, the residual analysis must give us the final
statistical basis for accepting or rejecting the proposed model.
128
since Ellison introduced them in 1947. Meyer and Wischmeier rela
tionships of the erosion subprocesses are frequently found in the
literature as part of the basic equations to describe soil erosion.
David and Beer (1975a, 1975b) proposed a sheet erosion model
where the basic imput was the overland flow depth (from a hydrologic
watershed model) and the rainfall intensity. This model considered
soil erosion from raindrop splash, stream banks and from impervious
surfaces by developing empirical equations for the detachment and
transport capacity due to raindrop splash and runoff. After the
model was calibrated, the results of the simulated daily, monthly and
annually suspended sediment loads were found to compare favorably
with the observed values.
Foster and Meyer (1972, 1975) presented a mathematical approach
for upland erosion by using fundamental erosion mechanics. Their
purpose was to be able to describe the erosion process at any time
and location in the watershed, including the prediction of deposition
of particles from overland flow. The basic equations of the model
were the sediment continuity equation expressed as
(2.69)
and the sediment loadflow detachment interrelationship
(2.70)
where
gs = sediment load [weight/(time unit width)]
Dp = detachment rate by flow [weight/(time unit area)]
Dj = detachment rate by rainfall [weight/(time unit area)]
349
measuring more parameters and their rate of change with respect to
time and space. However, this is necessary in order to maintain the
purpose of using a physically based model as much as possible and
preserves the validity of the model for a wider range of physical
conditions.
The significant difference in behavior of the curve correspond
ing to the 1.25 in./hr (32 mm/hr) rainfall intensity throughout the
presentation of some of the figures of this study may be due to the
high infiltration rates (21% to 40%) observed during those runs. For
the other rainfall intensity runs the infiltration rates were between
14% and 0.1%, with the infiltration rates decreasing as the bed slope
and rainfall intensity increased.
Instead of using the rainfall excess (rainfallinfiltration) to
describe the rainfall influence on the saltation length, the total
rainfall intensity is used because it is the only parameter that
accounts for the vertical momentum flux and may be used without ex
cessive computations. The influence of infiltration on the vertical
momentum flux is considered to be negligible when compared with the
rainfall term because the seepage velocity of water entering the soil
is significantly less than the raindrops impact velocity. Therefore,
the infiltration term may be neglected from the vertical momentum
flux term used in the saltation length without generating a signifi
cant error.
The proposed model may have an advantage over the use of
Yalin's bedload equation as suggested by Foster and Meyer (1972).
333
Table 5.16 PREDICTED su AND pVALUES
Run
su'
Value
Relative Error
(percent)
Probability
of Erosion, 
Perfect
Agreement
Equation
4.41
su
$
I
0.371
0.425
14.5
22.6
0.2632
V
0.400
0.381
4.59
13.0
0.6831
IX
0.334
0.359
7.56
22.5
0.8118
XIII
0.344
0.329
4.43
22.5
0.9289
XVII
0.318
0.314
1.12
7.13
0.9741
XXI
0.300
0.301
0.07
0.46
0.9921
II
0.424
0.400
5.74
13.8
0.5286
VI
0.366
0.340
7.23
33.1
0.8884
X
0.326
0.332
1.87
8.33
0.9302
XIV
0.303
0.319
5.19
25.3
0.9614
XVIII
0.292
0.301
3.09
19.8
0.9859
XXII
0.296
0.293
1.37
12.7
0.9950
III
0.399
0.380
4.97
14.1
0.6762
VII
0.330
0.326
1.23
5.93
0.9275
XI
0.309
0.309
0.01
0.16
0.9639
XV
0.290
0.291
0.21
1.33
0.9842
XIX
0.286
0.284
0.59
5.11
0.9928
XXIII
0.283
0.276
2.60
29.2
0.9976
IV
0.361
0.357
1.25
4.17
0.8064
VIII
0.315
0.318
0.83
4.06
0.9451
XII
0.300
0.291
3.28
24.9
0.9820
XVI
0.280
0.283
1.20
8.80
0.9895
XX
0.276
0.284
3.05
22.7
0.9935
XXIV
0.273
0.275
0.64
6.19
0.9979
380
Foster, G. R., and L. D. Meyer, "Transport of Soil Particles by
Shallow Flow," Trans, of the American Society of Agricultural
Engineers, 15, 99102, 1972a.
Foster, G. R., and L. D. Meyer, "A ClosedForm Soil Erosion Equation
for Upland Areas," in Sedimentation (H. A. Einstein Symposium),
H. W. Shen (Ed.), Colorado State University, Fort Collins,
Colorado, 1972b.
Foster, G. R., and L. D. Meyer, "Mathematical Simulation of Upland
Erosion by Fundamental Erosion Mechanics," in Proc. of the
Sediment Yield Workshop, USDA Sedimentation Laboratory, Oxford,
Mississippi, Nov. 2830, 1972, U. S. Dept, of Agriculture,
Agricultural Research Service ARSS40, 190207, 1975.
Foster, G. R., L. D. Meyer, and C. A. Onstad, "An Erosion Equation
Derived from Basic Equation Principles," Trans, of the American
Society of Agricultural Engineers, 20(4), 678682, 1977a.
Foster, G. R., L. D. Meyer, and C. A. Onstad, "A Runoff Erosivity
Factor and Variable Slope Length Exponents for Soil Loss
Estimates," Trans, of the American Society of Agricultural
Engineers, 20(4), 683687, 1977b.
Foster, G. R., and W. H. Wischmeier, "Evaluating Irregular Slopes for
Soil Loss Prediction," Trans. of the American Society of
Agricultural Engineers, 17, 305309, 1974.
Free, G. R., "Soil Movement by Raindrops," Agricultural Engineering,
33, 491494, 496, 1952.
Garde, R. J., and K. G. Ranga Raju, Mechanics of Sediment Transporta
tion and Alluvial Streams Problems, Wiley Eastern Limited, New
Delhi, India, 1977.
Ghadiri, H., and D. Payne, "Raindrop Impact Stress and the Breakdown
of Soil Crumbs," J. of Soil Science, 28, 247258, 1977.
Ghadiri, H., and D. Payne, "Raindrop Impact Stress," J. of Soil
Science, 32, 4149, 1981.
Gilley, J. E., and S. C. Finkner, "Estimating Soil Detachment Caused
by Raindrop Impact," Trans, of the American Society of Agri
cultural Engineers, 28(1), 140146, 1985.
Gilley, J. E., D. A. Woolhiser, and D. B. McWhorter, "Interrill Soil
Erosion: Part I. Development of Model Equations," Trans, of the
American Society of Agricultural Engineers, 28(1), 147153,
1159 V 1985a.
395
Vanoni, V. A. (Ed.), Sedimentation Engineering, ASCE Manual and
Reports of Engineering Practice, No. 54, American Society of
Civil Engineers, 1975.
Walker, P. H., J. Hutka, A. J. Moss, and P. I. A. Kinnell, "Use of a
Versatile Experimental System for Soil Erosion Studies," J. of
Soil Science Society of America, 41, 610612, 1977.
Walker, P. H., P. I. A. Kinnell, and P. Green, "Transport of a
Noncohesive Sandy Mixture in Rainfall and Runoff Experiments,"
J. of Soil Science Society of America, 42, 793801, 1978.
Warner, R. C., B. N. Wilson, B. J. Barfield, D. S. Logsdon, and P.
J. Nebgen, "A Hydrology and Sedimentology Watershed Model: Part
II. Users' Manual," Department of Agricultural Engineering,
University of Kentucky, Lexington, Kentucky, 1981.
Wentworth, W. E., "Rigorous Least Squares Adjustment: Application to
Some NonLinear Equations, I," J. of Chemical Education, 42(2),
96103, 1965.
Wenzel, H. G., Jr., and R. C. T. Wang, "The Mechanics of a Drop after
Striking a Stagnant Water Layer," Water Resources Center
Research Report No. 30, University of Illinois, Urbana
Champaign, Illinois, 1970.
White, C. M., "The Equilibrium of Grains on the Bed of a Stream,"
Proc. of the Royal Society of London, Series A, 174, 322339,
T94U:
Whitman, D., Program N0NLIN, Version 3.0, in BASIC and inspired by a
FORTRAN program by C. F. Wilcox, Box 683 Baker Lab., Cornell
Univ., Ithaca, New York, 14853, 1982.
Williams, J. R., "SedimentYield Prediction with Universal Equation
Using Runoff Energy Factor," in Proc. of the SedimentYield
Workshop, USDA Sedimentation Laboratory, Oxford, Mississippi,
Nov. 2830, 1972, U. S. Dept, of Agriculture, Agricultural
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Williams, J. R., "Testing the Modified Universal Soil Loss Equation,"
Proc. of the Workshop on Estimating Erosion and Sediment Yield
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Agriculture, Agricultural Reviews and Manuals, ARMW26,
157165, 1982.
Williams, J. R., and H. D. Berndt, "Sediment Yield Computed with
Universal Equation," J. of Hydraulics Division, ASCE, 98(Hyl2),
20872098, 1972.
123
watersheds showed poor correlation between observed and predicted
values. So, he indicated that the LfS factor was not adequate for
flat slopes and that the runoff energy factor and Pf factor needed
more attention in their evaluation.
Recently, Hahn et al. (1985), Mclssac et al. (1986), and
Schroeder (1987) studied the effect of the slope on soil loss from
disturbed soils and reclaimed soils from mined lands. They developed
linear regression equations in order to obtain the slope factor (S)
used in the USLE. A summary of other slope factor equations devel
oped by other researchers was presented by Mclssac et al. (1986).
Schroeder (1987) also indicated that the Kfvalues obtained from
the field were approximately 81% of the value suggested by the USLE
nomograph (Wischmeier et al., 1971) because the nomograph was based
on undisturbed soils and is not applicable to the spoil soils he
studied. He indicated that his resultant Kfvalues were generally
reacting to rather than being independent of the slope gradient.
There are more significant contributors to the USLE but reporting on
the use of this equation is not the purpose of the present work. The
reader is referred to other soil erosion sources (e.g., see Section
2.9).
2.7.3 Soil Erosion Models
The estimation and prediction of soil erosion is indeed not
simple. The use of the Universal Soil Loss Equation is not the only
approach researchers have followed. The literature presents many
studies in which new approaches were proposed. In many cases,
233
researchers sometimes have used the square root of the ratio as the
expression to be used to identify the dimensionless number (e.g.,
Froude number). Following this concept, the square root of the ratio
RMFn/0MF expressed by Equation 4.25 will be used in this study for
the evaluation of the rainfall effects on the saltation length. Now
Equation 4.25 can be expressed as
Knl
1/2
OMF
(I V+ h Ax)
g 1
1/2
cose
(4.26)
In this study, rainfall intensity and raindrop distribution are
unchanged. Therefore, the value of gj will be constant. Kilinc and
Richardson (1973) considered gj equal to unity and due to lack of ad
ditional data the unity value for gj will be used in this study too.
The value of g may change as the water flows downslope and the fully
turbulent flow is developed. Also the raindrop impacts affect the
value of g through the velocity profile of the overland flow. Yoon
(1970) presented some values for p as a function of Ref. His re
sults were scattered, but in all cases the value of g was closer to
unity for rainfall conditions than for no rainfall conditions. The
observed g values approached unity as the rainfall intensity in
creased when Ref < 2500 while for Ref > 2500, the g values were
basically constant at about g = 1.06 regardless of rainfall intensi
ty. Yoon's results were obtained for a physically smooth boundary.
However, they can be used to support the assumption of using g equal
to unity. The error introduced by using this value will be small
77
diameters falling from a height of 14 meters (45.9 ft) were used in
this study. The peak forces were found to occur within 13 ps to
21 ps of initial contact, ranged from 1.0 N (0.225 lb) to 3.8 N
(0.854 lb) and decreased to 0.5 N (0.112 lb) after approximately
100 ps. Their results showed that the times and magnitude of forces
and average pressures of impact were not well predicted from theory
using incompressible mechanics nor by the numerical technique applied
by Hwang et al. (1982). They indicated that the effects of compres
sibility, surface tension and viscosity were important in the impact
phenomenon and further research was needed.
Nearing and Bradford (1987) studied the effects of waterdrop
properties on the impact force of waterdrops falling from different
heights. The mean peak force was found to be a unique quadratic
function of waterdrop kinetic energy per waterdrop diameter square
(i.e., KEg/Dg kinetic energy per unit area) for all of the fall
heights and drop sizes tested. They were able to relate the peak
forces to drop shape oscillation during fall. By dimensional analy
sis the impact force was related to the time during impact by using
waterdrop properties like surface tension, viscosity, density, bulk
compressibility, and the waterdrop diameter and velocity.
2.5.1.5 Flow Velocity
The longitudinal velocity profile along the water depth of the
overland flow has been studied although not as extensively as needed.
The velocity profile usually depends on the flow regime, the flow
Reynolds number, the wall Reynolds number, and the roughness
189
Therefore, the evaluation of Areaj was neglected and the probabil
ity of erosion was obtained using only the value from the positive
side of the probability distribution (Area2 in Figure 3.7). For
overland flow with rainfall, it is expected that su and x0 will
increase giving lower n+ and n values. This means that n still
will be small enough to be neglected from the computation of the pro
bability of erosion. Consequently, the probability of erosion is ex
pressed here as
p = Area2 = ==
/2ir
(n2/2)
dn (3.40)
 1/2
(no + l/su^Tcrs^To^ ^su
3.4 Sediment Transport Equation
Now that the probability of erosion has been defined by Equa
tion 3.40, Equation 3.15 is rearranged in order to solve for the
volume of sediment transported per unit time and unit width, qs,
for overland flow with rainfall.
qs =
P i_
(1 p) C*
f_a_ I*JLZ.
CD de V
1/2
(3.41)
where
^1 ^3
C* =  = It has dimensions of the inverse area or inverse
a2 l1 length squared
Cj = function to account for the water depth, rainfall parame
ters, and bed slope on the saltation length
CHAPTER IV
DATA, PROCEDURES AND EVALUATION OF COEFFICIENTS
4.1 Introduction
The use of Equation 3.41 in the determination of the soil ero
sion due to overland flow with rainfall makes an evaluation of the C*
function and the su values necessary. Attempts to determine these
values are presented in this chapter.
4.2 Data
All the necessary physical data and information required to
confirm the basic hypothesis of this dissertation was obtained from
studies alreacjy available in the literature. From the literature
review made during this study it was found that most of the data
required to use this model was incomplete as usually is the case when
experimental work is not matched directly with theory. Most of the
data available are from studies in which the main purpose was to
determine an empirical equation for soil erosion based on simple
parameters of a general nature. These studies present data such as
general grain size distribution, or soil classification, rainfall
intensity, longitudinal slope, slope shape, soil cover protection,
and total runoff. Other hydraulic parameters like overland flow
velocity and depth, bed shear stress, and angle of repose of the
grain have been usually neglected. Other physically based models
available in the literature have used data not as the one required
here and, when some data was missing steady state or uniform flow
191
X
= V((Ys Y)ds) = 1/T'
Ycr Tcr^ ^Ys Y)ds)
t' = flow intensity function
Â¥q = flow intensity function for sloping bed surfaces
Si = angle of the path of the falling raindrops with respect to the
vertical axis
w = angle of detachment of the resultant force with respect to the
bed surface
H = detachment angle for horizontal bed
>5 = detachment angle for sloping bed
xxvi i
Figure 3.7. Evaluation of the Probability of Erosion (based on Chiu, 1972).
346
discharge, the flow velocity, the water depth, and the timemean bed
shear stress which can be obtained from a hydrologic model. Other
physical parameters which are needed are the specific weights of the
soil particles and of the transporting fluid, the effectivegrain
size, the angle of repose of the soil, the porosity or void ratio of
the bed surface and the bed slope. Rainfall parameters needed are
the rainfall intensity, the distribution of the raindrop sizes for a
given rainfall intensity, and the corresponding meanterminal fall
velocity of the raindrops.
Of all these parameters it appears that the most critical para
meters are the timemean bed shear stress, T0, the water depth, h,
the raindrop size distribution (required to obtain BÂ¡), and the
meanterminal fall velocity of the raindrops. The last two parame
ters are associated with the rainfall influence on the saltation
length. As mentioned in the literature review, the basic problem
with these parameters is that even though the rainfall intensity is
the same, the raindrop size distribution may be affected by the geo
graphic location, the seasonal variations, the temporal variation of
raindrop sizes within the storm event, the wind effects and the pre
sence of physical obstructions of the falling raindrop path. If
these conditions are known or controlled the evaluation of and
&I would not be that difficult.
The water depth and the bed shear stress are hydraulic parame
ters which are intimately related. The evaluation of each of them is
very critical. The water depth directly affects the saltation length
84
Yoon also concluded that for the test conditions of his study,
the effect of raindrop impact spacing on the velocity profile was
almost negligible. He also indicated that the mass flow rate of rain
fall was the dominant parameter affecting the local timemean velocity
profile. He reported that for a constant rainfall intensity the mo
mentum coefficient (8) decreased with increasing flow Reynolds number,
reaching an approximately constant value of 1.06 at Ref > 2,500.
The 8 values were found to also decrease with increasing rainfall
intensity and constant flow Reynolds number.
Kisi sel (1971) also used the hot film techniques to obtain the
velocity profile of overland flows over a plexiglass (physically
smooth) surface and over a physically rough boundary (k = 0.0283 in. =
0.719 mm) surface. For the flows over the smooth boundary with or
without rainfall, the velocity profile was subdivided into three
regions represented by the equations
Viscous sublayer U+ = Y+ (2.51a)
Buffer zone U+ = 5 n(Y+) 3.05 (2.51b)
Turbulent layer U+ = a An(Y+) + b (2.51c)
where
U
+
u
Y+ = yv*
V
a = 2.5 and b = 7.0
The data agreed well with those equations for both overland flow
with rainfall and with no rainfall. Perhaps the reason for this
agreement in the results was that the flows were always turbulent,
Ref > 1560 for no rainfall condition and Ref > 2670 for rainfall
conditions. However, data from runs with rainfall conditions fell
231
For overland flow with rainfall, the thickness of the viscous
sublayer can be of the same order of magnitude as the water depth,
(de < 6 < h). This means that raindrops may penetrate the viscous
sublayer, modify its flow form and then reach the soil surface. As
the Ref increases the flow becomes more turbulent and the viscous
sublayer thickness decreases <5 < dg. It is considered that when
the flow becomes a fully developed turbulent flow the viscous sublay
er thickness may be considered negligible or equal to zero. This
means that the soil surface is under the direct effect of the rough
ness generated turbulence. The rainfall generated turbulence is very
low compared to the turbulence generated by the roughness elements.
Therefore, the total effect can be considered equal to the effect of
turbulent flow without rainfall.
According to these considerations the saltation length should
increase s the overland flow intensity increases and reach a maximum
value when the flow becomes fully turbulent. In other words, the
saltation length becomes independent of the flow conditions and as
assumed by Chiu can only be a function of the grain size.
The longitudinal momentum flux due to overland flow at a cer
tain location along the longitudinal bed slope can be used as a
measurement for the flow conditions. The overland momentum flux
(OMF) is expressed as
OMF
dA = B p
a2
m
Bh
A;
(4.22)
where
Cmr = cropping management factor for rill area
Cs = = total sediment concentration in the water flow
Ctp = temperature correction factor in energy equation, Park et
al. (1983)
Ctm = temperature correction factor in momentum equation, Park
et al. (1983)
C = dimensionless friction coefficient
T
c* = a1a3/a2c1
Cf = DarcyWeisback friction factor
c1 = 8g Sf0,1/(1.481,8 *2)
c" = c'Nm1,8
D = Drag Coefficient
D50 = mean equivalent spherical raindropsize diameter for given
rainfall intensity
Dc = detachment capacity of flow
Dc0 = detachment capacity of flow at the toe of the sloping bed
De = equivalent spherical raindropsize diameter
Dp = soil detachment by runoff
DÂ¡ = soil detachment by rainfall
D' = ~V*S 1
du
D* *0df/tco
d = ds = grainsize diameter
d1Q = grain size with 10% of finer material
d31 = 9raln size w1th 31^ of: finer material
d35 = 9ran size with 35% of finer material
d50 = 9rain S1ze w1th 50^ f finer material
xv ii
340
It was mentioned before that the sediment concentration can af
fect the suvalues. Figure 5.16 shows the su required to obtain
perfect agreement versus the crosssectional mean sediment concentra
tion, Cs. As previously required in Section 4.3.2, the general
pattern is su decreasing as Cs increases. All curves except for
the 1.25 in./hr (32 mm/hr) rainfall curve have very similar forms
with the general tendency of su to decrease as rainfall intensity
or bed slope increases. This was previously related to the increase
of flow Reynolds number as rainfall intensity or bed slope increased.
The 1.25 in./hr (32 mm/hr) rainfall intensity shows some irregulari
ties which can be associated with the possible errors in the data
already discussed at the beginning of this chapter. Nevertheless,
the shape of this curve is not too far from the general shape of the
curves for the other three rainfall intensities.
Figure 5.17 presents the predicted suvalues, using Equa
tion 4.41, versus the sediment concentration. Here the pattern of
all curves approaches more closely a linear relationship between su
and An Cs. For sediment concentrations lower than 20%, there is a
tendency of each rainfall intensity curve to be independent of each
other. For higher concentrations, a single curve independent of
rainfall intensity, might be able to represent all suvalues.
The use of Equation 4.41 appears to be adequate for considering
the effects of Cs on su. Figure 5.17, with the predicted suvalues,
shows a remarkable similarity to the required suvalues required
for perfect agreement, Figure 5.16. This supports the use of
no
considered, they satisfactorily predicted the average trend of large
data sets taken as a whole. Their recommendations were suggested to
be viewed in a statistical sense and only within the conditions of
the data used in the study.
There are other studies in which existing sediment transport
equations were used and sometimes were modified in order to apply it
for soil erosion. Young and Mutchler (1969c) used the century old
DuBoys approach to express the soil erosion in small tillage channels
in which the soil was under dry conditions and then in wet condi
tions. They obtained coefficients of determination, r2, of 0.61
for the dry runs, r2 = 0.80 for the wet runs, and r2 = 0.69 for
the combined data. By adding a correction factor due to surface ten
sion effects the correlation improved to r2 = 0.85. This correc
tion factor was found to have a linear relationship (r2 = 0.782)
with the Weber number (We = dimensionless ratio of inertia forces
to surface tension forces) expressed as
We = p R' 0m2/r (2.58)
where r is the surface tension.
Komura (1976) used the KalinskeBrown equation to generate two
separate equations for soil erosion by overland flow on sloped beds.
One equation was developed for the region where the flow was laminar
and the other for the region with turbulent flow. An area coeffi
cient expressing the bare area in terms of the total surface area
and an erodibility coefficient to account for the various erosion
157
The equilibrium condition can be represented by the equation
Nd
(3.1)
where
Nd = number of particles deposited per unit time and unit bed
area
Ne = number of particles eroded per unit time and unit bed
area
This expression of the equilibrium transport condition was used
on the original Einstein's bed load transport equation (Einstein,
1950) for open channels and by other researchers who have modified
that equation including Chiu (1972) or in Christensen and Chiu
(1973). This study will also use the equilibrium transport condition
to develop the equations for the soil erosion by overland flow and
rainfall. In addition to the considerations already presented in
Section 3.1, new concepts and considerations which are necessary for
the application of this idea to overland flow will be presented
throughout this study whenever they are needed.
3.2.1 Evaluation of Ne
The number of particles eroded per unit time per unit bed area,
Ne, is defined by
N
e
e
Alde2ti
(3.2)
p = absolute probability that a particle is eroded (moved
from its original location)
p
Alde = bed area associated with one grain
ti = time to bring the grain out of the bed and into the
fl ow.
where
100
2.6 Rill and Gully Erosion
2.6.1 Rill Erosion
Rill development comes from the concentration of overland flow
toward lower elevations of the soil surface profile. The soil sur
face roughness (e.g., vegetation, soil particles and aggregates), the
bed slope, and soil erodibility characteristics (e.g., organic con
tent, particle size distribution, degree of aggregation and others)
will also affect the presence of rills on the soil surface (Ellison,
1947; Meyer et al., 1975a; Young and Onstad, 1978).
The effect of raindrop impacts in the rill area decreases due
to the increase of the water depth, but the concentration of overland
flow toward the rill allows the flowing water to effectively trans
port most of the soil particles which were detached in the interrill
area (Young and Wiersma, 1973). The flow in the rill may also allow
to erode soil particles from the rill area and, if the soil condi
tions are appropriate, rill erosion can be predominant too (Mosley,
1974; Young and Onstad, 1978).
Usually the rill grows upslope due to the erosion which occurs
at the headcut (local area of steep slope) at the upslope end of the
rill (Meyer et al., 1975a). However, sidewall and bottom erosion can
also occur if the walls and bottom materials are not erosion resis
tant and if the flowing water transport capacity has not been
reached. This also means that as the flow toward the rill increases
the rill erosion may also increase.
79
Machemehl (1968) continued Glass and Smerdon's research and
found that the von Ka'rmn constant was greater than 0.40 for no rain
fall conditions but smaller than 0.40 for flow with rainfall. He
concluded that the suspended sediment reduced significantly the value
of k, indicating that the mixing was less effective and that the sed
iment tended to suppress or dampen out the turbulence. His results
also indicated a reduction on the suspended load transport capacity
when rainfall was applied.
Previously, Nail (1966) also observed changes in the sediment
transportation in shallow flows, but he did not specify the magnitude
of these changes due to equipment and procedures limitations. Nail
explained the decrease of the flow velocity due to rainfall by con
sidering that rainfall increases the mass of the flow but does not
impart momentum in the direction of the flow. Therefore, the velo
city must decrease if momentum has to be conserved. This resulted in
a corresponding increase of water depth when rainfall is applied.
Unfortunately, all these studies were made under very high
rainfall intensities (greater than 20 in./hr, 508 mm/hr) over shallow
flows, which have water depths greater than those usually found in
overland flow, h < 0.1 ft (30 mm). In addition, the flow regime was
always turbulent (4000 < Ref < 87,000). These Reynolds numbers are
too high for the conditions usually found in overland flows.
Threadgill and Hermanson (1969) studied the rainfall effect on
the velocity distribution in a triangular shallow open channel for
physically rough and smooth surfaces. The Prandtlvon KaVman
18
oscillations are gradually damped and at terminal velocity their drop
shape observations showed that the drops attain equilibrium and have
an oblate shape. Comparison of the drop shape showed that the drops
obtained in the laboratory (still air conditions) were more oblate
than the drops of equivalent drop diameters in natural rain observed
by Jones (1959).
Some researchers have developed analytical approaches to des
cribe the raindrop size, shape, and falling speed. Spilhaus (1948)
assumed that a falling raindrop has an ellipsoidal shape. The sur
face tension effect was combined with the aerodynamic deformation of
the drop in order to maintain the steady shape and falling velocity.
His theoretical values partially agreed with Laws (1941) data, but
his approach was not able to describe the complex behavior of the
falling raindrop in air. McDonald (1954) presented a better analy
tical approach in which he concluded that under most conditions the
surface tension, the hydrostatic pressure and the external aerody
namic pressure were the three factors which had important roles in
producing the characteristic deformation of large raindrops.
Wenzel and Wang (1970) used a balance of forces approach to
stucty freely falling drops. That is, neglecting minor forces, they
considered the balance between the drag force, the buoyant force and
the gravitational force. Solving for the drag coefficient, CD, and
using data from Laws (1941) and Gunn and Kinzer (1949), they produced
diagrams for the drag coefficient of falling waterdrops in air. A
5.16 Required suValue Versus Measured Sediment
Concentration, C$ 341
5.17 Predicted suValues Versus Measured Sediment
Concentration, C$ 342
5.18 Relationship Between C$ and v+dg/v 344
B.l Correction Factor for Bed Shear Stress due to Longi
tudinal Slope for Kilinc and Richardson's Silty
Sand Material 363
B.2 Required suValues 370
xiv
119
results showed that ML = 0.30 for dry run conditions and
Ml = 0.15 for wet run conditions.
The slope gradient factor, S (dimensionless), was originally
presented by Smith and Wischmeier (1957) as a parabolic curve with
respect to the bed slope, S0 (in percent). The equation has only
been evaluated for bed slopes up to 18% (Singer and Blackard, 1982)
and the use of this equation on steeper slopes have indicated dis
agreement with the results obtained by most researchers.
Foster and Martin (1969) found that the soil loss decreased
with increasing bed slope after a certain bed slope value depending
on the soil properties and the slope length, was exceeded. This de
crease in soil loss was explained as a reduction of incoming rainfall
energy due to the decrease of the area being exposed to rainfall
against the expected increase of overland flow energy gradient due to
the steeper flow.
If the area under study has nonuniform slope (convex, concave
or complex topographic profiles) the total length of the slope should
be divided on segments of uniform slopes. Onstad et al. (1967), and
Foster and Wischmeier (1974) modified the USLE equation in order to
account for this variation and also for variations in soil type, and
cropping management. This modification was also provided with the
option of using the USLE to predict soil loss from individual storm
basis.
The cropping management factor, Cm, represents the ratio of
soil loss from a specific cropping or cover condition (e.g.,
experiences with our children, Raul Enrique and Mari Luz, I give my
deepest love, appreciation, and respect. I am also very grateful to
n\y parents and family for their patience and understanding during
this period of our lives.
I wish to express my deep appreciation to Dr. L. Martin, Dr. F.
Fagundo and their families for the friendship, guidance and help
which they have provided to me and my family during our stay in this
natural and beautiful city of Gainesville, Florida. I thank my fel
low graduate students and neighbors for their friendship and encour
age them to continue working hard to reach their goals.
v
232
= local longitudinal velocity (a timemean value)
m = crosssectional timemean value of u
8 = dimensionless momentumflux correction factor to account
for the variation of the velocity across the cross sec
tional area, Bh.
B =
Bh
u2dA
A=bh
Substituting the definition of discharge per unit width defined
as q = Umh into the equation gives
OMF = p 8 B q2/h (4.23)
Using Equation 4.21 and 4.23, the ratio between the component
of the rainfall momentum flux normal to the bed and the longitudinal
momentum flux due to overland flow may be expressed as
RMFn p 8Â¡ I Vt B Ax cos2e
OMF p 8 q2 B/h
(4.24)
Assuming that raindrop density and overland flow density are the
same, namely the density of fresh water, Equation 4.24 becomes
RMFn 8Â¡ I Vt h Ax cos20
Dimensionless numbers frequently used in fluid mechanics are
obtained from ratios between certain parameters with the same units.
In order to simplify the use of such dimensionless parameters,
381
Gilley, J. E., D. A. Woolhiser, and D. B. McWhorter, "Intern'll Soil
Erosion: Part II. Testing and Use of Model Equations," Trans.
of the American Society of Agricultural Engineers, 28(1), 154
159, 1981)5
Glass, L. J., and E. T. Smerdon, "Effect of Rainfall on the Velocity
Profile in Shallow Channel Flow," Trans, of the American Society
of Agricultural Engineers, 10, 330336, 1967.
Goldman, S. J., K. Jackson, and T. E. Bursztynsky, Erosion and
Sedimentation Control Handbook, McGrawHill Book Co., New York,
im;
Grace, R. A., and P. S. Eagleson, "Similarity Criteria in the Surface
Runoff Process," Report No. 77, Hydrodynamics Laboratory,
Department of Civil Engineering, Massachusetts Institute of
Technology, 1965.
Grace, R. A., and P. S. Eagleson, "The Modeling of Overland Flow,"
Water Resources Research, 2(3), 393403, 1966.
Graf, W. H., Hydraulics of Sediment Transport, Water Resources
Publications, Littleton, Colorado, 1984.
Gregory, J. M., and J. M. Steichen, "Soil Loss Equation for Single
Event Storms," International Symposium of Urban Storm Water
Management, University of Kentucky, Lexington, Kentucky, July
2427, 1978, 303307, 1978.
Gunn, R., and G. D. Kinzer, "The Terminal Velocity of Fall for Water
Droplets in Stagnant Air," J. of Meteorology, 6, 243248, 1949.
Haan, C. T., H. P. Johnson, and D. L. Brakensick (Eds.), Hydrologic
Modeling of Small Watersheds, American Society of Agricultural
Engineers, Monograph No. 5, St. Joseph, Michigan, 1982.
Hahn, D. T., W. C. Moldenhauer, and C. B. Roth, "Slope Gradient
Effect on Erosion of Reclaimed Soil," Trans, of the American
Society of Agricultural Engineers, 28(3), 805808, 1985.
Harlow, F. H., and J. P. Shannon, "Distortion of a Splashing Liquid
Drop," Science, 157, 547550, 1967a.
Harlow, F. H., and J. P. Shannon, "The Splash of a Liquid Drop," J.
of Applied Physics, 38(10), 38553866, 1967b.
Henderson, F. M., and R. A. Wooding, "Overland Flow and Groundwater
Flow from a Steady Rainfall of Finite Duration," J. of Geophysi
cal Research, 69(8), 15311540, 1964.
378
Ekern, P. C., "Problems of Raindrop Impact Erosion," Agricultural
Engineering, 34(1), 2325, 1953.
Ekern, P. C., Jr., and R. J. Muckenhirn, "Water Drop Impact as a
Force in Transporting Sand," Proc. of Soil Science Society of
America, 12, 441444, 1947.
Ellison, W. D. "Studies of Raindrop Erosion," Agricultural Engineer
ing, 25, 131136, 181182, 1944.
Ellison, W. D., "Some Effects of Raindrops and SurfaceFlow on Soil
Erosion and Infiltration," Trans, of the American Geophysical
Union, EOS, 26, 415429, 19W.
Ellison, W. D., "Soil Erosion Studies," Agricultural Engineering, 28,
145146, 197201, 245248, 297300, 349351, 353, 402405, 408,
442444, 450, 1947.
Ellison, W. D., "Splash Erosion in Pictures," J. of Soil and Water
Conservation, 5, 7173, 1950.
Ellison, W. D., and C. S. Slater, "Factors that Affect Surface
Sealing and Infiltration of Exposed Soil Surfaces," Agricultural
Engineering, 26, 156157, 162, 1945.
Elwell, H. A., and M. A. Stocking, "Raindrop Parameters for Soil Loss
Estimation in a Subtropical Climate," Agricultural Engineering
Research, 18, 169177, 1973.
Engelund, F., and E. Hansen, "A Monograph of Sediment Transport in
Alluvial Streams," Teknisk Forlag, Copenhagen, 1967.
Engman, E. T., "Roughness Coefficients for Routing Surface Runoff,"
J. of Irrigation and Drainage Engineering, ASCE, 112(IR1),
3953, 1986.
Epema, G. F., and H. T. Riezebos, "Drop Shape and Erosivity: Part I.
Experimental Set Up, Theory, and Measurement of Drop Shape,"
Earth Surface Processes and Landforms, 9, 567572, 1984.
Farmer, E. E., "Relative Detachability of Soil Particles by Simulated
Rainfall," Proc. of Soil Science Society of America, 37,
629633, 197T:
Farrell, D. A., W. C. Moldenhauer, and W. E. Larson, "Splash
Correction Factors for Soil Erosion Studies," Proc. of Soil
Science Society of America, 38, 510514, 1974.
65
Kisi sel1 s (1971) study showed the same trend as Yoon's results.
The increase in T0 and Cf with the increase in the rainfall intensity
was equally observed over both physically smooth and rough surfaces
and particularly for laminar and transitional flow Reynolds numbers.
In this study the Cf values also increased when the bed slope
increased. For the physically rough surface case the increase in
Cf was slightly larger than that observed for flows with rainfall
over the smooth surface. He indicated that for both surfaces, the
main factor affecting the friction factor values was found to be the
rainfall input.
The studies of Yoon (1970), Kisisel (1971), and Shen and Li
(1973) were conducted at nearly horizontal uniform slopes (S0 < 3%)
with a fixed bed. These are ideal conditions in which the dynamic
equation was found to be a useful tool to evaluate and Cf. Kilinc
and Richardson (1973) also used the dynamic equation for that purpose
too, but their study was conducted at steeper bed slopes {5.7% < SQ <
40%) and with a movable bed (silty sand).
Kilinc and Richardson obtained introducing Equations 2.18
and 2.24 into Equation 2.19. Their study considered steady state
conditions with B = 1, BÂ¡ = 1, q = (I f)x, assumed that the in
filtration rate (f) was constant along the bed slope and used h =
q/um to express the water depth. The solution for T0 at the
downstream end of the plot was obtained numerically using their ex
perimental data. These values which included rainfall effects
were found to be less than the calculated from Shen and Li's ex
pression for Cf, Equation 2.33, but greater than the t calculated
assuming uniform flow, Equation 2.28. The ^values were later
347
of the grains, the transport capacity of the flowing water and possi
ble protection of the bed surface from the raindrop impacts. Since
the water depths in the experiment used in this study were very
small, direct measurement of them was not possible. The reported
water depths correspond to the mean water depths obtained from the
unit discharge value divided by the longitudinal mean velocity. This
is a lump value to be used over an area in which the water depths are
on the order of magnitude of the roughness elements. This means that
some part of that area might have been exposed more to the rainfall
than others. Consequently, the detachment and transport of particles
can differ from the ones predicted.
In addition, this study only considered one grain size which
indicates that the C2~value is a dimensionless constant equal to
2.3. However, C2 = h^/de, will change as de changes. As indicated
in Section 4.3.1 the h^value is expected to be in the range of
de < h. < k. This initial water depth should account for the
roughness effects and the additional elevation the water surface has
to reach over the top of the grains in order to overcome the surface
tension and viscous forces and start flowing. For the silty sand
used by Kilinc and Richardson, this h^value corresponded to 2.3
de. Based on this value and following the h^ definition, the
dimensionless C2~value for any other bed material with different
effective grain size may be represented by the expression
308
The first assumption of the residual analysis assumes that the
expected value of the error of each point will approach zero. This
assumption deals with the model selection and whether additional
independent variables need to be included. If the first two steps
are satisfied this initial assumption on the residual analysis should
hold. Another way to verify this assumption is to use the residuals
of the ^values as the estimates of the error in each point. This
study used the natural logarithm of $ in the regression analysis.
Hence, the residual in this study corresponds to
Residual = In OBSERVED *n4 PREDICTED (5.14)
The expected value of the error can be obtained using the
expression
i=ND
Expected ^Residual )i 0.10605
Error = 111 = = 0.00442 (5.15)
ND 24
where ND represents the number of points used in the regression
analysis.
The mean value of the observed data is obtained (see also
Table 5.9) by:
i=ND
Data ^ OBSERVED^' 29.8771
Mean value = = = 1.2449
ND
24
(5.16)
6
Those modifications will be presented in Chapter III and IV where the
proposed approach is presented and developed using existing data from
the literature. In addition, an error analysis of the used data and
the predicted results is presented and discussed in Chapter V. The
possible advantages and limitations of the proposed method are
discussed. Other discussions and conclusions are also presented.
Throughout this work the unit system used is the English Sys
tem. However, in the review of other studies related to soil erosion
the unit systems used in those studies are used. The proper unit
conversion is presented in such cases. See also Appendix C for the
conversion factors for units between the English System and the SI
System.
252
>1 1
2
10'2 1
2
103 1
LEGEND
Rainfall
inThr
mm/hr
1.25
32
o
2.25
57

3.65
93

4.60
117
Cbsrved
1 t i i i i i
ill! 1,1,1
i o
111
1 O 2
10'2
1 2
10'1
XF'
1
10
10
Figure 4.10. Comparison of Observed and Predicted
Dimensionless Sediment Transport.
81
was also indicated that for a constant rainfall intensity, the flow
velocity retardation was more pronounced near the water surface than
near the bed boundary. This flow retardation was observed to in
crease as the rainfall intensity increased.
Yoon divided the velocity profile in two portions and each of
them was described by a different equation. These two portions are
shown in Figure 2.4 which presents a typical velocity profile and
shear stress distribution for the cases of flow without rainfall and
with rainfall for the same flow Reynolds number in each case. For
the lower portion the equation to represent the velocity profile was
based on the Prandtl's mixing length hypothesis or von KaVma'n's
similarity hypothesis. This portion of the velocity profile
(0 < y < ymax) was expressed by the equation
'max
 u
=  [An(l /T
y/y ) + /l y/y ]
max max
(2.49)
where umax is the maximum local timemean velocity at a distance ymax
from the bed surface. The upper portion of the velocity profile used
the mixing length theory with the assumption of linear shear stress
distribution which gives the following equation for the velocity
profile in the upper portion (ymax < y < h)
umax "u
'*s
i y /y y ma x
=  [ (/hy + D') An(l )+ /yy ]
K max ^"~ymax + D max
(2.50)
where
16
understand the erosion process due to rainfall. Raindrop character
istics important in soil erosion are the drop mass, size, shape, and
their terminal velocity. Falling raindrops in air are not completely
spherical, but researchers have referred to an equivalent spherical
diameter De based on the actual mass of the raindrop to discuss the
variation in size between waterdrops.
Laws (1941) presented velocity measurements of waterdrops with
sizes ranging from 1 mm (0.039 in.) to 6 mm (0.236 in.) in diameter
falling through still air from heights of 0.5 m (1.64 ft) to 20 m
(65.6 ft). He also reported a few measurements of raindrop veloci
ties in order to compare with earlier observations. Lav/s' measuring
techniques consisted of a high speed photographic system, used to
measure the drop velocity and the flour pellet method to determine
the drop size. Laws' results showed that the waterdrops attained a
terminal velocity after falling a certain height. The height re
quired to reach terminal velocity increased as the drop size in
creased for drop sizes of about 4 mm (0.157 in.) or less. Beyond
that drop size the required height gradually decreased as the drop
size increased. The variations in the drop shape and the consequent
change in the friction resistance through the drop falling stage were
related to that reduction of the required height to reach terminal
velocity. Nevertheless, the terminal velocity always increased as
the drop size (i.e., drop mass) increased.
Later, Gunn and Kinzer (1949) presented what appears to be the
most accurate fall velocity measurements available. Using electronic
294
decreases the value of the limiting water depth. Let the rainfall
intensity increase more than the values used by Kilinc and Richard
son. Then, Equation 4.29 would become valid for a 5.7% bed slope at
rainfall intensities higher than 9.25 in./hr (235 mm/hr) and for
rainfall intensities higher than 8.60 in./hr (219 mm/hr) on the 40%
bed slope. That is to say, at those rainfall intensities or higher
Equation 4.28 can be used in the saltation length evaluation at any
given water depth.
However, the validity of the model at water depths very close
to zero is also limited to the initial assumption presented in Chap
ter III, in which the average length traveled by a particle is said
to be represented by Equation 3.8
Â£* = L_ (3.8)
1 p
The condition for using this equation is based on uniform steady
flow. However, the assumption that the rate of change of the water
depth with respect to the longitudinal distance is very small may al
low the use of the equation for overland flow with rainfall. There
fore, the rate of change in saltation length with respect to the lon
gitudinal distance will also be smaller and the values of the proba
bility of erosion (p) and the average saltation length (4) in a given
longitudinal increment could be assumed constant over that increment
section. Therefore, the use of Equation 4.28 for the description of
the saltation length has to satisfy those original conditions too.
The data from Kilinc and Richardson show very small mean water depth
368
Table B.4 PREDICTED su AND pVALUES
Run
su
Probability
of Erosion, 
Perfect
Agreement
Equation
4.41
Relative Error
(percent)
I
0.380
0.442
16.14
0.2830
V
0.408
0.398
2.51
0.7088
IX
0.348
0.375
7.84
0.8457
XIII
0.359
0.344
4.13
0.9497
XVII
0.340
0.329
3.31
0.9903
XXI
0.315
0.315
0.03
0.9992
II
0.437
0.416
4.69
0.5387
VI
0.377
0.355
5.77
0.8925
X
0.340
0.347
1.92
0.9395
XIV
0.321
0.334
4.05
0.9711
XVIII
0.316
0.316
0.12
0.9943
XXII
0.308
0.307
0.43
0.9994
III
0.411
0.396
3.64
0.6786
VII
0.340
0.341
0.29
0.9282
XI
0.322
0.323
0.38
0.9675
XV
0.306
0.305
0.23
0.9876
XIX
0.307
0.298
2.83
0.9970
XXIII
0.293
0.290
1.14
0.9997
IV
0.369
0.372
0.76
0.8025
VIII
0.324
0.333
2.53
0.9445
XII
0.312
0.305
2.25
0.9829
XVI
0.294
0.300
1.32
0.9914
XX
0.296
0.299
0.99
0.9971
XXIV
0.283
0.289
2.06
0.9997
10
the flow discharge has enough velocity in itself to produce erosive
bed shear stresses. However, the raindrop impacts and the corres
ponding splashes are very significant in any area before the rills
are generated and the areas between the rills. These two areas are
usually referred to as the interrill areas.
Now, it is necessary to know the extent of the rainfall effects
on the soil erosion process. Young and Wiersma (1973) studied the
relative importance of raindrop impact and flowing water to the ero
sion process. This was accomplished by determining the source and
mode of sediment transport on a laboratory plot under conditions of
normal rainfall energy and greatly reduced rainfall energy. They
found that decreasing the rainfall impact energy by 89% without
reducing rainfall intensity, the soil losses decreased by 90% or
more. It was thus demonstrated that the impact energy of raindrops
is the major agent in soil detachment. For all three soils studied
80% to 85% of the soil loss originating in the interrill area was
transported to a rill before leaving the plot. Thus, it was indica
ted that the transport of detached particles from the plot was accom
plished mainly by flow in the rills.
From that study, Mutchler and Young (1975) found that the soil
carried along by the splash energy was only 10% to 17% of the soil
loss from interrill areas to rills. The remainder of the loss to
rills was carried in the thin surface flow which without raindrop
impacts carried little, if any, soil. Therefore, the conclusion was
57
cf
24 + 27.162 I
0.407
Re,
(2.33)
The transition to an apparent turbulent regime has been report
ed at flow Reynolds numbers from 100 to 1000. The higher values usu
ally corresponded to the smooth boundaries. Shen and Li (1973) used
Ref = 900 as the maximum Reynolds number in the laminar flow regime
over smooth surface while Yoon (1970) established this maximum Ref
in the range of 1000.
Savat (1977) has presented a summary of other maximum laminar
Reynolds numbers reported in the literature. He considered that a
turbulent flow was believed to prevail when Ref > 1000, the transi
tional flow occurred when Ref 500 and a laminar flow when Ref < 250.
Savat also indicated that the maximum laminar Reynolds number changed
with changes on the bed slope as seen in other investigations.
For the turbulent flow regime, there are many proposed rela
tionships to use. The Blasius equation
cf =
0.233
Ref0'25
(2.34)
(Woolhiser, 1975) can be used for smooth boundary flows without rain
fall and a Reynolds number less than about 30,000. Robertson et al.
(1966) used the same type of equation to express the friction factors
for three different rough boundaries under rainfall conditions. Un
fortunately, for the flow Reynolds number range tested in the study
(400 < Ref < 4500), the coefficients of their equations changed for
each rough surface studied.
134
The potential soil detachment rate due to rainfall, DÂ¡, was
expressed as
Dj = a! I2 [1 (hw/3D50)] (1 Cg) (1 Cc) hw < 3D50
(2.76)
Dj = 0 hw > 3D50
where
aj = coefficient which depends on soil characteristics
hw = water depth plus the loose soil depth
Cg = ground density cover factor
Cc = canopy density cover factor
The amount of soil detached by surface runoff was determined by com
paring the total sediment transport capacity to the total available
amount of loose soil. The model can present estimates of erosion (or
deposition) and sediment yield distributions in time and space on a
storm event basis. The particle size distribution of the eroded
material could also be estimated. Other papers in which some of Li's
approaches were used or tested with specific examples are Li et al.
(1976, 1977) and Simons et al. (1978).
Lane and Shirley (1982) and Rose et al. (1983b, 1983c) have
also derived analytical solutions for sediment yield. They have used
the kinematic wave approach to obtain the hydraulic parameters re
quired in their models and the sediment continuity equation as their
basic governing equation. Lane and Shirley (1982) separated the ero
sion process into interrill and rill erosion, and integrated with
323
due to changes in bed slope, water depth and rainfall intensity.
Considering these three effects, the saltation length appears to be
more dependent on the flow conditions than the originally assumed
by Chiu. However, the assumption of a saltation length being
independent of the flow conditions may be considered valid in fully
developed turbulent flow as found in rivers and major drainage
canals. However, overland flow usually does not favor this flow
condition, and the saltation length must be expected to change with
the flow conditions.
Figure 5.13 shows the saltation length ratio following the same
approach used in Figure 5.11. In Figure 5.13, the surface curve pre
sents a steeper gradient between the maximum and minimum values of
the saltation length ratio, point IV and point XXI, respectively.
The bed slope effects which were not completely visible in Figure
5.11, for the points with almost the same water depth, are more dis
tinctive. Points I (S0 = 5.7%), XIV (S0 = 20%), XIX (SQ = 30%)
and XXIV (S0 = 40%) have corresponding dimensionless saltation
length ratios of 0.341, 0.294, 0.270 and 0.211, which indicates that
the saltation length decreases as the bed slope decreases even though
the rainfall intensity is increasing.
After carefully studying the saltation length predicted by this
model and studying the data from Kilinc and Richardson, it is found
that Equation 4.47 does not directly account for the saltation length
of grains being transported through the air and later returning to
164
affected, either. This happens because the rainfall intensity is
assumed constant. Consequently, the raindrops will have the same
impact effects over the area with the same water depth.
It is necessary to mention that overland flow can also occur on
steep slopes. This means that bed slope effects usually neglected on
the saltation length of particles in rivers (bed slopes are usually
less than 2%) can become significant for overland flow with or with
out rainfall. Some of the bed slope effects can be related to the
decrease of the force required to detach grains from the bed surface
as the bed slope increases. However, further considerations must be
discussed before a full description of the saltation length can be
finally presented.
As a summary, the average saltation length will be considered
constant at each depth increment section with constant rainfall in
tensity, but it will change its value from section to section. Now
Equation 3.8 can be applied for the specific location along the slope
under study given that the average saltation length and the proba
bility of erosion definitions are based on the specific local condi
tions. The change of these variables along the longitudinal distance
of the flow will be accounted for at each specific location along the
longitudinal distance. The bed slope, the water depth and the rain
fall intensity at that given location will be used to obtain the cor
responding saltation length and probability of erosion of the sec
ti on.
Introducing Equation 3.8 into Equation 3.7, the number of par
ticles deposited per unit time and unit bed area is expressed as
q (1p)
d a2 de3*
(3.9)
372
Surface
Tension
N/m
lb/ft
1 lb/ft = 14.594 N/m
Volumetric
Discharge
m3/s
ft3/sec
1 m3/s = 35.315 ft3/sec
Vol. Discharge
per unit width
m3/(s m)
ft3/(secft)
1 m3/(s*m) = 10.764
ft3/(secft)
Energy per
unit area
J/m2
(ft 1b)/ft2
1(ft 1b)/ft2 = 14.594
J/m2
Energy per
unit area
per unit time
J/(m2s)
(ft*lb)/(ft2* sec)
l(ft*lb)/(ft2* sec) =
14.594 J/(m2*s)
Momentum per
unit area
N/m2
lb/ft2
1 lb/ft2 = 47.88 N/m2
Momentum per
unit area
per unit time
M/(m2 s)
lb/(ft2,sec)
lb/(ft2'sec) = 47.88
N/(m2's)
Weight per
unit time
per unit width
N/(s*m)
1b/(sec ft)
1 lb/(sec*ft) = 14.594
N/(s *m)
I certify that I have read this study and that in niy opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Bent A. Christensen, Chairman
Professor of Civil Engineering
I certify that I have read this study and that in ny opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
I certify that I have read this study and that in rny opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the
degree of Doctor of Philosophy.
Erik R. Lindgren
Professor of Engineering Sciences
This dissertation was submitted to the Graduate Faculty of the
College of Engineering and to the Graduate School and was accepted as
partial fulfillment of the requirements for the degree of Doctor of
Philosophy.
December 1987
lluJhju/~ Ci'
Dean, College of Engineering
Dean, Graduate School
2
stucty is only concerned with the soil erosion due to rainfall impact
and flowing water. Most of the observations presented here are based
on water. Nevertheless, wind erosion is recognized as an eroding
agent which has a significant effect in the shaping of the earth's
surface.
The soil erosion problem has been widely studied by many scien
tists and engineers, but it is still not well understood. The main
reason is that soil erosion is a very complex process and involves so
many variables that it is practically impossible to measure the
influence of all of them in one study.
In addition, the scale of this process is so small that it
makes it practically impossible to measure the variables in an accur
ate manner even with today's sophisticated equipment. Researchers
cannot produce a model with dimensions longer than those of the pro
totype because not all of the dimensions or variables can be modeled
to the same scale (e.g., waterdrops larger than 7 mm are unstable
(Blanchard, 1950) and the terminal velocity of the waterdrops is
influenced by surface tension while in a larger model surface tension
may be relatively negligible). Using the prototype size requires
equipment with almost microscopic dimensions in order to measure flow
parameters at different locations rather than average values with
larger instruments. Therefore, scaling and instrumentation are
problems that soil erosion researchers have to deal with.
Soil erosion is also an unsteady and stochastic process. Des
cribing it requires knowledge of how the variables change with time.
In most of the studies, the investigators have selected some
125
might give an error greater than 40% for a single storm event on a
small watershed.
Rowlison and Martin (1971) presented a rational model to des
cribe erosion on sloped beds. They considered that the maximum ero
sion rate at a given location on the slope plane was a function of
the bed slope, the unit weight of the soil and the depth of the water
flow. The maximum erosion rate was considered to be the smallest
value between the transport rate and the detachment rate. This cri
terion is based on a given soil, simulated rainfall intensity, and
flow conditions. When the possible range of values are consolidated
into an isometric figure, using water depth and bed slope as indepen
dent variables, the maximum erosion rate values form a surface parti
ally defined by the transport rate and the remaining part is defined
by the detachment rate (see Figure 2.5). This maximum erosion rate
surface was controlled in part by the detachment of soil particles
due to raindrop impact and in part by the transport capacity of both
runoff and raindrop impact. Laboratory results from previous studies
by these researchers satisfied the concept presented with the surface
of maximum erosion rate.
Meyer and Wischmeier (1969) developed a mathematical model of
soil erosion by water based on Ellison's (1947) approach of consider
ing the soil erosion as "a process of detachment and transportation
of soil materials by erosive agents" (rainfall and runoff). Empiri
cal equations, based on available information related to the basic
190
Given the definition of the effective grain size, de, Equa
tion 3.41 represents the total rate of sediment transport. The equa
tion is also based on the assumption of a cohesionless bed material
with a minimum effective grainsize dg of about 50um. This means
that predominantly very fine materials such as fine silts and clays
are not included in Equation 3.41. Also for the case of a soil bed
material with high concentration of fine material, the cohesion be
tween the grains is increased. This will definitely affect the de
tachment and transportation of the soil grains assumed in this study.
Therefore, the Equation 3.41 cannot be used to consider the transport
of fine soils.
To complete the use of Equation 3.41 in overland flow with
rainfall conditions, the C+ function and the su values have to be
evaluated. The next chapter presents an attempt made to obtain those
values. The validity of this physicallybased model with its assump
tions is later discussed in Chapter V.
56
For laminar flow over rough boundaries Cf can be represented by
(2.30)
where K is a parameter related to the characteristics of the bed
surface and can be as large as 40,000 for dense turf (Wool hi ser,
1975).
For overland flow with rainfall, the raindrop impacts increase
the K factor and it has been represented by the expression
(2.31)
where KQ is the K value without rainfall and ar and bp are empirical
coefficients. Tables with typical values for KQ, ar, and br are pre
sented in Woolhiser's (1975) study. Woolhiser also indicated that
for smooth boundaries (K0 = 24) the raindrop impact effect is
important, but it becomes insignificant for vegetated surfaces (K0
> 3000).
Izzard (1944) was among the first researchers to use this
approach in his study of runoff over rough paved plots. His results
suggested the following equation
= 27(0.21 I4/3 + 1)
(2.32)
Shen and Li (1973), using data from various studies of overland
flow with rainfall over smooth boundaries, proposed the following
equation if Ref < 900.
183
o4c u't2[1 + (ui/5t)2]
(3.28)
Let
2
o =/ u! = standard deviation of the velocity fluctuation
u t
ut
n = = normalized velocity fluctuation
au
u
su = = = dimensionless standard deviation of the velocity
ut fluctuation
For overland flow without rainfall, su values may be low or
at least on the order of magnitude found in open channels, i.e.,
su 0.15 or su = 0.164 as obtained indirectly from Einstein and
ElSamni (1949) lift studies by Christensen (1965). But, impacting
raindrops increase the turbulence of the overland flow and the su
values may also increase significantly.
Substituting these definitions into Equation 3.27 and 3.28, and
then dividing Equation 3.27 by Equation 3.28 yields that
1 + S,
(3.29)
Then substituting this expression for  T0 j into the erosion
criterion inequality 3.26 results in
T
0
(1 + n Su)2
1
>
A2
A1 Eh
T
crs
(X/  T0  ) + cot
(3.30)
Christensen (1973) showed that the (X/ frQ  ) ratio is equiva
lent to (X/  to  ) ratio and it is a function of the roughness to
 Eroded soil characteristics and properties.
 Nutrient and pollutant migration, etc.
4
Most of the studies found in the literature seem to describe
empirical approaches where, from observed data, equations are devel
oped using some kind of a regression analysis. One disadvantage of
this approach is that the developed equations are only valid for the
specific conditions that existed during the observations. Extrapola
tion, of course, cannot be recommended, and even interpolation may
have its problems. The other disadvantage of using empirical ap
proaches is that the resulting empirical relationships can only pre
dict the mean values of the observations for a certain condition and
any information relating to fluctuations is lost. Some of the varia
bles upon which many of these studies are based are the kinetic ener
gy or momentum of raindrops, the rainfall intensity, some of the soil
properties, vegetative cover, slope and length of the ground surface
and the conservation practice used in order to prevent soil loss.
Attempts have been made to explain soil erosion by the basic
physical laws. However, the results are usually limited to very nar
row parts of the whole process. Continued research is definitely
needed. Attempts to physically describe the effect of raindrops and
sheet flow at the same time have been only partially successful and
the literature on this topic is quite limited.
Attempts of using existing sediment transport equations origin
ally developed for water courses have been made, but the results have
not been satisfactory because the boundary conditions are different.
ACKNOWLEDGMENTS
I want to express n\y most sincere appreciation to Dr. Bent A.
Christensen, committee chairman, for the direction, advice and assis
tance which he has given to me throughout my graduate studies at the
University of Florida. His knowledge, moral support and patient
guidance helped me to complete this study.
Thanks are due to Dr. E. R. Lindgren for his teaching lessons
in fluid mechanics and for serving on the supervisory committee.
Thanks are also extended to Dr. L. H. Motz for serving on the super
visory committee.
Special thanks are extended to the University of Puerto Rico,
Mayaguez Campus, for providing me the opportunity to improve my know
ledge, securing me a leave of absence and financial support through
out try studies. Thanks are also extended to the Center for Instruc
tional Research Computing Activities at the University of Florida for
the use of their facilities.
Thanks are extended to Gail Luparello and Irma Smith for the
quality typing, and also to Katarzyna Piercey and her husband, Dr. R.
Piercey, for their beautiful drawings.
To ny wife, Maria del Carmen, whose loving support and encour
agement has always inspired me and allowed me to have beautiful
iv
85
below the U+ = Y+ curve indicating that there was an increase on the
boundary shear stress compared to the runs with no rainfall condi
tions. Also in the outer region, the coefficients of Equation 2.51c
seemed to be smaller than those for flow without rainfall. The cause
of this effect was considered to be the increased rate of vertical
momentum transfer from the impacting raindrops.
For the rough boundary without rainfall the velocity profile was
divided into two portions. The first one, from relative depths (y/h)
of 0.015 to 0.15 was described by the equation
lL,y r (2.52a)
max = 4.5 Â£og(y/h) + 1.5
v*
while the rest of the velocity profile was described by
max
 u
= 6.0 Aog(y/h)
(2.52b)
where EÂ¡max is the local maximum time mean velocity measured in the
profile. For the range of flows tested in this study, flows with
higher Reynolds numbers agreed more with Equation 2.52b than the ones
with the low flow Reynolds numbers (Ref = 2545). Kisi sel indicated
that the reason for this was that equations like Equations 2.52a and
2.52b are known to be valid for fully turbulent flows at high Reynolds
numbers.
For flows over the rough boundary with rainfall he used the
equation
= 0.3 3.1 Aog(y/h) + 1.5
(2.53)
74
impact stress. This was contrary to the results obtained by Palmer
(1963, 1965).
Cruse and Larson (1977) altered the soil strength in three dif
ferent ways, i.e. by bulk density changes, pore pressure changes and
addition of polyvinyl alcohol (PVA) providing an intergranular bound
ing, to test the relationship between a single raindrop impact (4.8
mm (0.189 in.) raindrops falling from a height of 1.77 m (5.8 ft))
and the soil strength parameters. The amount of soil splashed was
closely correlated with the soils shear strength determined from the
triaxial compression test. The coefficient of determination (r2)
of 0.86 was obtained when a second degree polynomial regression was
used. The general tendency of the results was that the soil strength
increased with an increase of the bulk density or increase in PVA
content or decrease in pore pressure. This study appears to be the
first one in which pore pressure, bulk density and soil strength were
monitored throughout the soil detachment process.
AlDurrah and Bradford (1981) also modified the bulk density
and the pore pressure of the soil in order to investigate how the
soil detachment by single raindrop impact (raindrops size of 3.0 mm,
4.6 mm and 5.6 mm or 0.118 in., 0.181 in., and 0.22 in., respective
ly) is related to the soils shear strength (as measured in a standard
fallcone device). The linear relationship
WS = a: + a2 KEd/t (2.46)
where
312
for this study are too limited to be expected to fully reproduce the
straight line of the normal distribution. However, the points do
show a general agreement with the straight line behavior. This sug
gests that the errors might be normally distributed.
The last assumption is that the errors must be statistically
independent and hence, uncorrelated. This assumption deals with ser
ial observations (i.e., time series) and the relation of the error at
one point to the error at the next point in the series. The serial
sequences which this data could have are the rainfall intensity and
bed slope. For each bed slope sequence there are four data points,
and for each rainfall intensity there are six data points. There are
not enough numbers to draw a conclusion about the serial dependence
of the error with the rainfall intensity or the bed slope. In the
previous discussion of the possible error in the data, however, there
were indications that the errors in Cs tended to increase as the
rainfall intensity decreased or as the bed slope increased. Also,
there was a discussion of the increase of the possible error in Vq as
the bed slope increased. Since the discussion centered on the data
error and not on the model's error, this discussion cannot be used
here to consider the independence of the residuals in the model.
Kilinc and Richardson reported that there was no significant
change in the sediment concentration, Cs, with respect to time.
Hence, the values they reported and used in their regression analysis
were values averaged over the onehour run.
397
Wolman, M. G., "The Natural Channel of Brandywine Creek, Pennsylva
nia," U. S. Geological Survey, Prof. Paper 371, 1955.
Woo, DC, and E. F. Brater, "Spatially Varied Flow from Controlled
Rainfall," J. of Hydraulics Division, ASCE, 88(Hy6), 3156,
1962.
Wooding, R. A., "A Hydraulic Model for the CatchmentStream Problem:
I. KinematicWave Theory," J. of Hydrology, Amsterdam, 3(3),
254267, 1965a.
Wooding, R. A., "A Hydraulic Model for the CatchmentStream Problem:
II. Numerical Solutions," J. of Hydrology, Amsterdam, 3(3),
268282, 1965b.
Wooding, R. A., "A Hydraulic Model for the CatchmentStream Problem:
III. Comparison with Runoff Observations," J. of Hydrology,
Amsterdam, 4(1), 2137, 1966.
Woolhiser, D. A., "Overland Flow on a Converging Surface," Trans, of
the American Society of Agricultural Engineers, 12, 460462,
1565:
Woolhiser, D. A., "Simulation of Unsteady Overland Flow," Unsteady
Flow in Open Channel, K. Mahwood and V. Yevjevich (Eds.), Water
Resources Publication, Fort Collins, Colorado, 1975.
Woolhiser, D. A., and P. H. Blinco, "Watershed Sediment YieldA
Stochastic Approach," in Proc. of the Sediment Yield Workshop,
USDA Sedimentation Laboratory, Oxford, Mississippi, Nov. 2830,
1972, U. S. Dept, of Agriculture, Agricultural Research Service
ARSS40, 264273, 1975.
Woolhiser, D. A., and P. Todorovic, "A Stochastic Model of Sediment
Yield for Ephemeral Streams," U. S. Department of Agriculture,
Miscellaneous Publication No. 1275, 295308, 1975.
Wright, A. C., "A PhysicallyBased Model of the Dispersion of Splash
Droplets Ejected from a Water Drop Impact," Earth Surface
Processes and Landforms, 11, 351367, 1986.
Yair, A., "Theoretical Considerations on the Evolution of Convex
Hi 11 slopes," Z. fur Geomorph. N. F. Suppl. Bd. 18, 19, 1973.
Yalin, M. S., "An Expression of BedLoad Transportation," J. of the
Hydraulics Division, ASCE, 89(Hy3), 221250, 1963.
Yalin, M. S., Mechanics of Sediment Transport, 2nd Edition,
Pergamon Press, Ltd., Oxford, England, 1977.
276
the bed slope increases. Also, the asymptotic behavior of ^ no
longer exists when v' is used.
The rainfall effects on the erosion process are seen more
clearly using ^instead of V. In Figure 5.4, the pattern of the
curve corresponding to the 1.25 in./hr (32 mm/hr) rainfall intensity
approaches the general pattern of the other three rainfall intensity
curves more closely than it does in Figure 5.3, where y1 is used.
Each curve in Figure 5.4 shows an almost straight section between the
10% bed slope and the 40% bed slope. The slope of the straight
section of the curve is nearly the same for each of the four rainfall
intensities. In Figure 5.3, the values of H'1 for the 1.25 in./hr (32
mm/hr) rainfall intensity curve show a pattern which appears to be
slightly different from the pattern of the other curves. The notable
difference is the absence of an apparent asymptotic value of V as
the bed slope exceeds the 20% value.
Figure 5.4 also shows that for small bed slopes the rainfall
effects significantly change the ^values, however, as the bed slope
increases, these effects are outweighed by the slope effects. For
the 5.7% bed slope, changes about 70% when the rainfall intensity
changes from 1.25 in./hr (32 mm/hr) to 4.60 in./hr (117 mm/hr), while
the change is about 36% for the 40% bed slope.
Figure 5.5 presents the ^values calculated from the data ob
tained in Kilinc and Richardson's study. This figure is similar to
241
in order to obtain an estimate of the suvalues needed in this
study.
Other problems which have to be considered are the effect of
the relative roughness (k/h) and small flow Reynolds numbers on su.
In shallow overland flow the relative roughness approaches to unity
which means that roughness elements can obstruct the water flow and
change the local timemean velocity distribution and the turbulent
velocity fluctuations used on the evaluation of su. The values of
the flow Reynolds number commonly encountered in overland flow indi
cate that the flow is not always fully developed turbulent flow. In
some cases the flow may be nearly laminar corresponding to a zero
suvalue. However, the effect of the roughness might be enough to
keep the velocity fluctuations present, even under viscous flow con
ditions.
When it rains, the level of turbulence is increased and su
must be expected to increase. The turbulence in overland flow with
or without rainfall has been studied by Yoon (1970), Kisisel (1971)
and Shahabian (1977). As mentioned before, their velocity measure
ments were mostly made at fairly great distances from the bed surface
(y/h > 0.05). ExtraDolation of Yoon results presented in Figures 36
and 37 of his dissertation show that for a rainfall intensity of 15
in./hr (380 mm/hr) the order of magnitude of su can be as high as
0.5 for a flow Reynolds number of 450 and as low as 0.10 for a flow
Reynolds number of 4000. For rainfall intensities of 1.25 and 3.75
in./hr (32 and 95 mm/hr) the values of su were around 0.40 for the
306
Table 5.14 SUMMARY OF DATA POINTS WITH LARGEST ERROR ON THE PREDICTED
MODEL SOLUTION
Run
Slope
(percent)
Rainfall
Intensity
(in./hr)(mm/hr)
Reasons for Possible Error
I
5.7
1.23
(32)
y value is high, Figures 4.5, 5.2, 5.3,
and 5.4
VI
10
2.25
(57)
$ value has large error or y' value is
high, Figures 5.2, 5.5
IX
15
1.25
(32)
4*' value is high, Figures 4.5, 5.2, 5.3,
and 5.4
XII
15
4.60
(117)
41 value is high, Figures 4.5, 5.2
0
XIII
20
1.25
(32)
Secondary effects of Run IX by using
least squares method without any
correction in data.
XIV
20
2.25
(57)
$ value has large error, Figure 5.5
XVIII
30
2.25
(57)
o value has large error, Figure 5.5
or r' value is high, Figures 4.5,
5.2 6
XX
30
4.60
(117)
*e' value is low, Figures 5.2, 5.3, 5.4
XXIII
40
3.65
(93)
p = 1 (discussed later in Section 5.5)
29
Contemporary to Mutchler's works, Hobbs presented another ser
ies of articles about waterdrop splash characteristics. Hobbs and
Kezweeny (1967) measured the number of droplets produced by the im
pact of a waterdrop on a water surface and the electric charge of
these droplets. The number of droplets produced was a function of
the fall distance of the waterdrop. A fall distance of 10 cm (3.94
in.) or less did not produce any splash, and for fall distances up
to 200 cm (78.7 in.) the number of droplets was found to increase
linearly with the fall distance. Mutchler's results cannot be com
pared with the results obtained by Hobbs and Kezweeny because the
latter study did not test fall distances higher than 200 cm (78.7
in.). Consequently, no terminal velocity of the waterdrop was
reached in this latter study. It should also be mentioned that the
latter study reported that nearly all of the spray droplets carried a
negative charge and for the range of fall distances used it appeared
that the fall distance had little effect on the charges carried by
the spray droplets.
Hobbs and Osheroff (1967) and Macklin and Hobbs (1969) also
studied the effect of the water layer depth on the waterdrop splash
but their major interest was the study of the Rayleigh jet produced
by the returning (converging) fluid filling the crater created by the
waterdrop impact.
The waterdrop splash has also been studied analytically using
the NavierStokes equations. Each study has established its assump
tions and boundaries to the problem resulting in simplified equations
which are solved by numerical analysis and computers.
239
m r
tanel
C2 cos e
tarty
C ( YBi 1 vt cos2fi)
(4.34)
Equation 4.34 is written in the English Unit System. It also
shows that the coefficients to be evaluated by experimental data are
C2, ^3 ^4 ar)d
4.3.2 su Considerations
The relative turbulence intensity, or dimensionless standard
deviation of the velocity fluctuations near the solid boundary,
7 1/2
su = C(ut ) ^/f ls a ver> important parameter in this stochastic
model. For a given flow condition, the probability of erosion can be
obtained from the corresponding know values of t tcr$, and su.
The literature shows for turbulent flow with no rainfall that
typical suvalues are around 0.15 and 0.20, (e.g., McQuivey and
Richardson, 1969; Lyles and Woodruff, 1972, and others). Other stud
ies indicate that su has a constant value (Taylor, 1932; Laufer,
1950; Christensen, 1965, and others) and the suggested values are on
the same range of su values. The equivalent sand roughness of the
bed is the parameter most frequently associated changes in suvalues.
Usually an increase in bed roughness indicates an increase in su.
244
Figure 4.8 shows the predicted suvalues using C* = 2.2 x 107
ft2. The general trend is that each curve representing a constant
rainfall intensity may be approximated by an equation similar to
Equation 4.35. However, not all of the constant rainfall intensity
curves follow the same pattern. The curve corresponding to the rain
fall intensity of 1.25 in./hr (32 mm/hr) shows a completely opposite
behavior to the rest of the curves. It is expected that all curves
must have a similar behavior in order to say that the model has some
validity. The reason for the strange behavior of the curve corres
ponding to 1.25 in./hr is the use of C* as a constant in spite of C*
now being expressed by Equation 4.34.
Also, although not mentioned before, the concentration of sedi
ment, Cs, in the water may reduce the value of su. When the sed
iment concentration in the fluid increases, the apparent viscosity
and density of the mixture increase while the turbulence, i.e., the
velocity fluctuations, may decrease. As a result of these conditions
the value of su must decrease as the sediment concentration in
creases. The effect of sediment concentration may be minimal for
small concentrations of sediments (Cs < 2.5%). However, as the con
centration increases the effect becomes more significant. Figure 4.9
shows the predicted suvalues versus the sediment concentration,
using C* = 2.2 x 1C7 ft2 (constant). The curve corresponding to
the rainfall intensity of 1.25 in./hr (32 mm/hr) shows a general
trend which is unrealistic. The function defining must correct
that. The expected solution should show the same general pattern for
all rainfall intensities in Figure 4.9.
124
improved versions of some of them are available too. Some of them
are discussed here due to their contribution to this work or to
describe the soil erosion process.
Bennett (1974) presented some concepts for mathematical model
ing of soil erosion and the needs for research on each of the two
phases of the erosion process: the upland slope (overland flow in
its three forms: sheet, rills, and gullies) and the lowland stream
phase (a wellestablished water course flowing through the valley).
The need for explaining the effects of unsteadiness and flow nonuni
formities in the eroded soil transport was also indicated by this
author as well as for the need for sensitivity analyses of the
models.
For each phase of the erosion process, Bennett presented and
discussed the basic equations (i.e., conservation of mass and momen
tum equations of the flow and conservation of mass equation for the
sediment) in which a model should be constructed. He also discussed
the resulting limitations which necessarily must be accepted when
simplifications of these equations are used. These simplifications
can reduce the work and time required by the users to obtain a solu
tion but the loss of accuracy and representation of the real condi
tion in the watershed was recognized too. He presented the compari
son that, although estimates obtained from regression of annual sedi
ment discharge based on annual flow might be expected to be within
20% of the observed values even a wellcalibrated digital model might
117
combined effects or raindrop impact and turbulence of runoff to
transport soil particles from a field.
Usually the R factor is indicated on an annual basis with units
of El/year, but the R values show a seasonal distribution during the
year (Wischmeier, 1959; Wischmeier and Smith, 1965; Mitchell and
Bubenzer, 1980). Researchers have found that the R factor mey be
overestimated at rainfall intensities above 3.0 in./hr (76.2 mm/hr)
because the median raindrop size does not continuously increase at
higher rainfall intensities, (e.g., Hudson, 1963; Carter et al.,
1974). Current recommendations (Wischmeier and Smith, 1978) propose
a limit of 3.0 in./hr (76.2 mm/hr) be used in the computation of
energy per unit rainfall, Equation 2.6, and that a maximum limit of
2.5 in./hr (63.5 mm/hr) be placed on the 130 value for the evalu
ation of R. There should also be indicated that the R factor is the
major term in the USLE and in the soil loss estimation.
The soil erodibility factor, Kf, represents the erodibility
of a particular soil with respect to other soils when all the remain
ing soil erosion factors are kept constant. The Kf factor of a
soil represents the amount of soil loss per unit area per El unit.
Initial measurements were carried out in studies like Olson and
Wischmeier (1963) and later on by Barnett et al. (1965), Barnett and
Rogers (1966), Wischmeier and Mannering (1969), Barnett (1976), Young
and Mutchler (1977), Meyer and Harmon (1984) and others. Additional
measurements have been made at many locations worldwide. The U.S.
Soil Conservation Service has evaluated the Kf factor for most of
302
grain Reynolds number, v^dg/v, used in the prediction of su, is
directly related to through v*.
Data point XXIII does not show any irregularity in the evalua
tion of $ or Likewise the value of to 1S not indicating any
thing out of the general pattern established by other points. Study
of the remaining variables used in the evaluation of Equation 4.37
does not reveal the possible source of any measurement error. How
ever, in studying the model's behavior the presence of an error is
indicated by the magnitude of the probability of erosion and the mag
nitude of the error of su. Because there is an entire section ded
icated to the behavior of su and p, the error of data point XXIII
is not discussed here. It is fully discussed in Section 5.5 of this
chapter. At the moment, it indicates that this error is associated
with the sensitivity of the model as the probability of erosion
approaches unity.
Predicted $values are plotted versus the observed values of $
and in Figure 5.8. None of the 24 data points have an error larger
than 1.56 times the standard error of the estimate, oerr = 0.1873.
There are 15 data points out of the total 24 points with an error
less than the standard error of the estimate. These 15 points repre
sent 62.5% of the sample. If the error of the predicted values fol
lows a normal distribution the expected number of observations should
be around 68.3% at a distance of one standard deviation. For two
standard deviations the percentage is around 95.5%. Since the ob
served ratios of the data used are very similar to the ones predicted
151
By combining most of the factors discussed in this chapter,
researchers have proposed descriptions and sometimes models for the
slope evolution or the geomorphology of this planet (e.g., Horton,
1945; Smith and Bretherton, 1972; Yair, 1973; Mosley, 1973; Kirkby
and Kirkby, 1974; Moss and Walker, 1978, and others). Such descrip
tions and the possible predictions that may be derived from them is
the final goal of many researchers who study the landscape forming
aspects of the soil erosion process. This will allow for better
agricultural management practices and the preservation of the natural
resources for the future generations to come.
The soil erosion process has not yet been fully understood.
More research is still needed. The present work is expected to pro
vide an additional contribution to the description of the process and
help other researchers with a stochastic representation of the soil
erosion by overland flow with rainfall.
2.9 Recent Books on Soil Erosion
An excellent book on soil erosion, the book edited by Kirkby
and Morgan (1980), presents nine chapters about different topics in
soil erosion. Each chapter includes an extensive list of references
published up to 1978. Another good general reference is the chapter
written by Foster for the book edited by Haan et al. (1982). Other
recent books associated with soil erosion are authored by Toy (1977),
Morgan (1979), Holy (1980), Morgan (1986) and Goldman et al. (1986).
In addition, some of the proceedings from conferences sponsored by
the Soil Conservation Society of America, the Soil Science Society of
236
be used for the given rainfall intensity and bed slope conditions.
It should be clear that this initial depth will change as the rain
fall conditions and bed slope change. It is expected that at least
the initial depth should approach to zero as the rainfall intensity
increases because the available energy to detach and transport soil
particles generally increases as rainfall intensity increases (i.e.,
Wischmeier and Smith, 1958, and others). However, care must be taken
for the case of high rainfall intensities, e.g., I > 5 in./hr = 127
mm/hr. As indicated in Section 2.3, the larger raindrop sizes in any
rainfall intensity are limited to about 6 mm due to the unstability
of larger raindrop sizes. Consequently, the increase in rainfall in
tensity may result in a change of the raindrop size distribution, a
reduction of the mean raindrop size, and a decrease of the rainfall
kinetic energy of those high rainfall intensities (Mihara, 1951; Hud
son, 1963; Carter et al., 1974).
The C2 constant in Equation 4.30 must have a positive value.
Consequently, the C4 constant must also be positive if Inequalities
4.29 or 4.30 are to be satisfied.
Considering that Equation 4.27 accounts for the influence of
rainfall and water flow, the function f(I) must take into considera
tion the rainfall effects near the noflow condition. Overland flow
inertia forces are not important in f(I), but gravitational forces
are. The only forces detaching and moving the soil particles comes
from the raindrop momentum flux. The resisting force is the buoyant
weight of the soil particles. Therefore, f(I) may be expressed as
80
velocity profile was used but the water depth used for the hydraulic
radius in the evaluation of the shear velocity was substituted by the
resistance radius, R, defined as
where
B
J_ (h*)2/3dx*
A
Jo
(2.48)
h' = local water depth of the flow at x' = x'
x' = the distance measured across the flow from its bank
They indicated that Equation 2.47 is independent of the roughness of
the boundary but their results showed that ap and bp were different
from the expected 2.5 value and that the value of these parameters
changed with surface roughness, rainfall intensity, and location
along the triangular crosssectional area. Again the results were
obtained in the turbulent range (calculated Ref > 23,000) and the
water depth was between 2 in. (50 mm) and 6 in. (152 mm). Therefore,
this study could mostly apply in gullies rather than in overland
flow.
Yoon's (1970) doctoral dissertation presented for the first
time the most complete local timemean velocity profiles along the
vertical direction for sheet flow due to rainfall over a physically
smooth surface using hot film measurement techniques. As mentioned
before for a constant Ref, the overall effect of rainfall on the
timemean velocity profile was to reduce the mean flow velocity and
hence to increase the water depth. These characteristics were more
pronounced for Ref < 1500 and as rainfall intensity increased. It
264
Table 5.4
ESTIMATED
RELATIVE ERROR OF $> ,
Vg', AND
RELATED VARIABLES
Relative Error
(percent)
Aq$
Transport
A $
A V
Ay'
Run
Â£
^s
Rate
$
'V
Â¥e
I
11.2
2.81
26.2
20
21.0
V
18.7
2.07
33.7
20
21.9
IX
8.26
2.45
23.3
20
23.2
XIII
8.08
2.23
23.1
20
24.9
XVII
10.3
2.61
25.3
20
30.0
XXI
11.6
2.02
26.6
20
40.8
II
8.45
1.54
23.5
20
21.0
VI
14.5
1.13
29.5
20
21.9
X
10.8
1.43
25.8
20
23.2
XIV
12.1
1.63
27.1
20
24.9
XVIII
22.4
2.09
37.4
20
30.0
XXII
19.9
1.69
34.9
20
40.8
III
5.42
1.12
20.4
20
21.0
VII
11.6
0.93
26.6
20
21.9
XI
13.5
1.17
28.5
20
23.2
XV
12.2
1.26
27.2
20
24.9
XIX
9.87
1.75
24.9
20
30.0
XXIII
6.25
1.45
21.3
20
40.8
IV
8.00
0.81
23.0
20
21.0
VIII
5.23
0.83
20.2
20
21.9
XII
5.35
0.94
20.4
20
23.2
XVI
10.6
1.13
25.6
20
24.9
XX
10.7
1.64
25.7
20
30.0
XXIV
7.90
1.34
22.9
20
40.8
Maximum
relative error assuming maximum
absolute
errors in any direction
positive
or negative
with respect to the
observed
value.
Relative error of transport rate qs based on DuBoys' formula and using
the critical shear stress of the sloping bed based on Equation 3.21 in
stead of Equation 3.22.
280
show that the data points with the steeper slopes were more suscepti
ble to errors than were the data points with flatter slopes. The
exceptions to the latter generalization are the data points corres
ponding to 10% bed slope. However, this can be explained by the fact
that these data points follow the previously stated conditions of the
relative error increasing as rainfall intensity decreased.
These two general trends of the relative error may be explained
as follows. First, for a constant slope, the increase in rainfall
intensity will correspond to an increase in the water discharge and
water depth. Since the mean water depth is of the same order of mag
nitude as the roughness elements, the sediment transporting capacity
of the water is directly affected. The number of obstructions that a
grain may meet during a saltation jump increases as water depth de
creases. If the flow discharge is small (i.e., lower rainfall inten
sity) the particle will be more dependent of raindrop impacts in or
der to be transported over or around the roughness elements.
The locations of raindrop impacts on the considered surface
area are completely random. Since the discharge and the water depth
increase as the rainfall increases, the sediment transport becomes
better established. This happens since the crosssectional area of
sections where a grain can pass over or around the roughness elements
increases and the sediment transport becomes more dependent on over
land flow characteristics and less dependent on the raindrop impacts.
Second, the rainfall intensity is considered to be constant
while the bed slope is changed. It is assumed that the infiltration
292
significance, the hypothesis of C4 = 0 cannot be rejected. How
ever, if C4 is zero, the rainfall effects on the saltation length
would not be accounted for. Apparently, based on statistical means,
the influence of rainfall effects on the saltation length can be ful
ly neglected and the model may still predict reasonably well the ero
sion process observed by Kilinc and Richardson. The elimination of
C4 from the least squares method would also produce new estimates
of the remaining coefficients.
Nevertheless, physical considerations have to prevail over the
statistical conditions of the model. It is recognized that rainfall
helps to move the grains when the water depth is very shallow, and
flowing water alone cannot move it (h < C2 cos e). The Kilinc and
Richardson data partially support the use of C4 t 0, but more
data are needed in order to prove it by statistical means.
Another condition which needs to be statistically defined is
the validity of the equation at no water depth (Inequality 4.29) or
the establishment of the limiting water depth for each rainfall
intensity (Inequality 4.30). Using Table 4.2 for the values of f(I),
the inequalities were tested. The presumption of Inequality 4.29 has
to be eliminated for the estimated coefficient values of C2 = 2.3
and C4 = 8.8. Consequently, the limiting water depths in which the
equations of the proposed model can be applied are obtained from
Inequality 4.30. These depths are summarized in Table 5.11.
The limiting water depth gradually decreases as the bed slope
increases. However, the increase in rainfall intensity significantly
386
Mantz, P. A., "Incipient Transport of Fine Grains and Flakes by
FluidsExtended Shields Diagram," J. of the Hydraulics Division,
ASCE, 103(Hy6), 601615, 1977.
Mazurak, A. P., and P. N. Mosher, "Detachment of Soil Particles in
Simulated Rainfall," Proc. of Soil Science Society of America,
32, 716719, 1968.
Mazurak, A. P., and P. N. Mosher, "Detachment of Soil Aggregates by
Simulated Rainfall," Proc. of Soil Science Society of America,
34, 798800, 1970.
McDonald, J. E., "The Shape and Aerodynamics of Large Raindrops," Jj_
of Meteorology, 11, 478494, 1954.
McGregor, K. C., and C. K. Mutchler, "Status of the R Factor in
Northern Mississippi," in Soil Erosion; Prediction and Control,
Proc. of a National Conference on Soil Erosion, May 2426, 1976,
Purdue University, West Lafayette, Indiana, Soil Conservation
Society of America, 135142, 1977.
McIntyre, D. S., "Permeability Measurements of Soil Crusts Formed by
Raindrops Impact," Soil Science, 85, 185189, 1958a.
McIntyre, D. S., "Soil Splash and the Formation of Surface Crust by
Raindrop Impact," Soil Science, 85, 261266, 1958b.
Mclsaac, G. F., J. K. Mitchell, and M. C. Hirschi, "Slope Gradient
Effects on Soil Loss from Disturbed Lands," American Society of
Agricultural Engineers Paper No. 862037, presented at the 1986
Summer Meeting ASAE, 1986.
Meyer, L. D., "Simulation of Rainfall for Soil Erosion Research,"
Trans, of the American Society of Agricultural Engineers, 8(1),
6365,' T965'.
Meyer, L. D., "How Rain Intensity Affects Interrill Erosion," Trans.
of the American Society of Agricultural Engineers, 24(6),
T4741475, 1981.
Meyer, L. D., G. R. Foster, and S. Nikolov, "Effect of Flow Rate and
Canopy on Rill Erosion," Trans, of American Society Agricultural
Engineers, 18(5), 905911, 1975a.
Meyer, L. D., G. R. Foster, and M. J. M. Romkens, "Source of Soil
Eroded by Water from Upland Slopes," in Proc. of the Sediment
Yield Workshop, USDA Sedimentation Laboratory, Oxford, Missis
sippi, Nov. 2830, 1972, U. S. Dept, of Agriculture, Agricultur
al Research Service ARSS40, 177189, 1975b.
98
increasing raindrop sizes did not result in increases of the sediment
discharge. The splash erosion was found to be higher only at the
upper part of the plot where the rainfall intensity or rainfall kin
etic energy were the highest.
Moss et al. (1979) presented a very descriptive study of the
raindrop induced transportation effects in shallow flows over poorly
sorted sand with bed slopes ranging between 0.1% and 30%. They
called rainflow transportation of solids to the shallow water flow
which is impacted by raindrops and allow the bed particles to be dis
turbed or removed from their resting position. This kind of flow was
found to be effective on slopes at least as low as 0.1% could operate
in flows less than a millimeter deep and exist in both subcritical
and supercritical flows.
Their results showed that this flow moved particles downslope
much more effectively than did the raindrop splash. Also, for the
case of low slopes, the raindrops impacting the shallow water tend to
suppress channel formation and to promote the sheet flow. This rain
flow transportation was found to be relatively insensitive to the
slope changes. This is contrary to the sheet flow with no rainfall
for which the transport capacity is very low for nearly horizontal
slopes. As the slope increases the transport capacity increases very
rapidly reaching significant values at slopes of 20% to 30%. Conse
quently raindropinduced rain transport is relatively important at
low slopes but may become overshadowed by the transporting action of
overland flow on medium and steep slopes.
114
2.7.2 The Universal Soil Loss Equation
There are many empirical soil loss equations available in the
literature but the most widely used in the United States is the
Universal Soil Loss Equation (USLE) expressed in English units as
E = R Kf Lf S Cm Pf (2.61)
where
E = rate of soil loss [tons/(acreyr)]
R = rainfall erosivity factor [El unit/year] =
i30 Â£[(916 + 331 Aog Ij)(Ij tj)]
J t
kf = soil erosivity factor [tons/acre'EI]
ml
Lf = slope length factor = (X/72.6)
S = slope gradient factor = 0.065 + 0.0045 SQ + 0.0065 SQ2
Cm = cropping management factor
Pf = erosion control practice factor
subscript j = specific storm increment with n number of incre
ments and a maximum 30 minute rainfall intensity of
I3q (in./hr)
tj = time period of the specific storm increment (hr)
Mj_ = exponential coefficient based on the bed slope, S0
This equation was originally developed by Wischmeier and Smith
(1965, 1978) as a simple method to predict annual soil loss from
intern11 and rill erosion plots. The equation was originally ob
tained from statistical analyses of more than 8000 plotyear data
186
Let n = n0 correspond to the probability of erosion when
t0 = Tcrs. Then, according to Equations 3.25 and 3.30 the Shields
entrainment function Eh can be expressed as
A* cot 4>+ U/T0)
1 + s,
(1 + n0 su)
(3.35)
Substituting this expression for Eh into the definition of 82
from Equation 3.31 yields
B2 =
(! + nQ su)2
1 + s,
1/2
1 + no su
(1 + su2)1/2
(3.36)
Introducing this Equation 3.36 into the criterion of erosion of
Equations 3.33 and 3.34 yields
1 + n s
n = n > 22.
+ ~
su
Tcrs ^ + su ^
To (1 + su >
1/2
1_
s,
and
n = n < 
1 + no su
Tcrs ^ + su ^
2iU/2
T (1 + S 2)
0 1 u '
which may be written
\ (no + 1/su)("Tcrs
n ^no + l/su^\rs
1
CNJ
iH
l/su
(3.37)
/T )l/2 _
' o'
1/SU
(3.38)
Chiu used n0 = 3.09, corresponding to the probability of ero
sion p = 103 when tq =
To maintain consistency between
178
advantage of this Equation 3.23 is that the direction of the buoyant
force will be known for any flow condition (e.g., uniform or nonuni
form flow), and a transition between the approaches used to develop
Equations 3.21 and 3.22 can also be explained.
Equation 3.23 may partially explain the discrepancy on the ex
perimental data collected by Olivier (1967), Fernandez Luque and van
Beek (1976), and others with respect to the values obtained from
Equation 3.22 because none of these studies had real uniform flow
conditions. In Olivier's study, the scarce data, lack of additional
information and the magnitude of error reported do not allow any fur
ther comment except that the flow was nonuniform with very irregular
water surface. However, the study by Fernandez Luque and van Beek
shows good agreement of the data with Equation 3.21. This study was
made in a rectangular closed conduit. For closed conduits the angle
Â£ is obtained from the piezometric head line slope with respect to
the horizontal. Unfortunately, this angle l cannot be directly ob
tained from their data, but assuming constant velocity head along the
closed conduit and the low timemean bed shear stress values (i.e.,
1.1 < ^Qcr < 2.7) used in the study, allows the consideration that
the angle S was more closer to 0 (zero) than to the bed slope angles
considered (i.e., 12, 18, and 22). Therefore, the good agreement
of the results from Equation 3.21 and possibly Equation 3.23 may be
justified.
2.6 Rill and Gully Erosion 100
2.6.1 Rill Erosion 100
2.6.2 Gully Erosion 102
2.7 Soil Erosion Estimates and Prediction .... 103
2.7.1 Use of Existing Stream Sediment Trans
port Equations in Overland Flow . . 104
2.7.2 The Universal Soil Loss Equation ... 114
2.7.3 Soil Erosion Models 123
2.8 Soil Characteristics and Slope Effects in
Soil Erosion 144
2.8.1 Soil Characteristics 144
2.8.2 Slope Gradient Effects 148
2.9 Recent Books on Soil Erosion 151
III. DEVELOPMENT OF THE SOIL EROSION EQUATION 153
3.1 General Purpose and Considerations 153
3.1.1 Basic Considerations and Assumptions 153
3.1.2 Major Considerations and Assumptions 154
3.2 Equilibrium Transport Condition 156
3.2.1 Evaluation of Ne 157
3.2.2 Evaluation of Nd 159
3.2.3 Evaluation of 159
3.2.4 Evaluation of Average Saltation
Length, i 165
3.2.5 General Equilibrium Transport Equation. 166
3.3 Relationships between Probability of Erosion
and Bed Shear Stress 169
3.3.1 Criterion for Erosion 169
3.3.2 Probability of Erosion 187
3.4 Sediment Transport Equation 189
IV. DATA, PROCEDURES, AND COEFFICIENTS EVALUATION ... 191
4.1 Introduction 191
4.2 Data 191
4.2.1 Soil Properties 192
4.2.2 Effective Grain Size Evaluation .... 199
4.2.3 Drag Coefficient Evaluation 200
4.2.4 Angle of Repose Evaluation 202
4.2.5 Shields Entrainment Function and
Critical Shear Stress Evaluations . 203
4.2.6 TimeMean Bed Shear Stress Evaluation 204
4.2.7 Grain Reynolds Number Evaluation . 204
4.2.8 Dimensionless Sediment Transport
Parameter 204
4.2.8.1 Evaluation of $ 204
4.2.8.2 Evaluation of'i'' and 4' . 205
4.2.9 Sediment Transport Data in Diagrams . 206
vii
278
Figures 4.2 through 4.5, Figure 5.3, and Figure 5.4. However, Figure
5.5 is presented here in order to support the statement that the sed
iment transport still increases as the bed slope increases (i.e.,
supports the use of V over y'). The curves are all smooth and show
the same general pattern. Some of the individual $values may have
some error in them but the similar pattern of each curve tends to
indicate that most of the irregularities observed in the $ versus ^
diagrams of this study may have come from errors of y' rather than
errors of $.
From the values of estimated errors of $ presented in Table 5.4
the average value of the estimated relative error of $ is 26%. The
estimated relative error of individual data points $values show more
fluctuations than the errors of the y'values and do not show any
e
dependency of the bed slope or rainfall intensity. This randomness
of the error of $ comes from the information provided for the sedi
ment concentration, Cs, already presented in Table 5.2. However,
it is possible that there is a general trend of the error of$val
ues. The average value of the estimated relative errors in each set
of data points with the same rainfall intensity are presented in Ta
ble 5.6. Similarly, for the sets of data points with the same bed
slope, the average value of the estimated relative errors on each set
are presented in Table 5.7.
These average values show the tendency that data points of the
two lower rainfall intensities were more susceptible to error than
data points of the two higher intensities. These average values also
53
C = (8g/Cf)0,5 = Chezy's coefficient
Cf = Darcy Weisbach's friction factor based on pipe diameter
Eagleson (1970) reported that experimental data from Horton
(1938) showed that the bk value was about 2.0 for natural surfaces,
and that further studies had supported that value for different kinds
of surfaces (e.g., vegetated surfaces, clipped grass, and tar and
gravel). The fluctuations of the bk exponent had been associated
with roughness effects. Usually an increase in roughness is associa
ted with the increase of the water depth which means a decrease of
the exponent's value. Muzik's (1974) results showed that bk was
exactly 1.66 = 5/3 for a galvanized sheet metal surface treated with
a diluted solution of hydrocloric acid to change the nonwetting
metal surface into a wetting surface.
The value of a^ is obtained based on known values of NM or C.
Woolhiser (1975), Lane and Shirley (1975), Podmore and Huggins
(1980), Engman (1986) and others have presented tables of typical
values for Manning's and Chezy's C coefficients which can be
used in overland flow studies.
The method of characteristics is frequently used to solve the
kinematic wave equations because it only has a single characteristic
relation to solve, namely,
k m
(2.26)
since
(2.27)
8
reported strain values occurred when the water depth was about one
drop diameter. Also, when the water depth was 20 mm (0.787 in.), the
stressstrain relationship was found to be about the same as without
the water layer. For a deeper water layer, the waterdrop impact
effects became negligible. He also reported that v/hen the depth of
the water layer was three times the drop diameter or more, the soil
loss was very small.
If the water layer is moving, then there is an additional ac
tive force creating soil erosion. Such overland flows usually have
very small depths and their mean velocities are not high enough to
produce erosive bed shear stresses. But the turbulence due to the
raindrop impacts does make the increased detachment and transport of
particles possible. This is because the raindrop impacts increase
the energy and the momentum transfer in overland flows.
In any rainfall event, the overland flow will be present after
the topsoil is saturated and the rainfall intensity exceeds the
infiltration capacity of the soil. Usually the depth and the veloci
ty of the sheet flow increase as the water moves downslope because
more rainfall is accumulated as the contributing area increases.
This flow tends to move towards microchannels in which the accelera
ting water tends to increase the scouring action forming rills or
small channels that usually grow in dimensions in the downstream di
rection. The rills may carry the water into bigger channels called
gullies where the now sedimentladen flow continues its erosive ac
tion. Finally, the water will reach a continuously flowing stream
39
The presence of vegetative cover or any manmade cover over the
soil will reduce the splash erosion because this cover will absorb
most of the raindrop energy (Mihara, 1951; Free, 1952; Young and
Wiersma, 1973, and others). The cover prevents surface sealing; con
sequently, the infiltration is not drastically reduced and the rate
of runoff is diminished. Free (1952) reported that the presence of
straw mulch reduced splash loss to about onefiftieth (1/50) of that
from bare soil and sheet flow losses to onethird (1/3).
Osborn (1954) indicated that, in addition to the already men
tioned soil and rainfall characteristics, the land use management and
conservation practices also affect the splash erosion. Other soil
characteristics not mentioned before were also related to this ero
sion process.
2.5 Overland Flow Erosion
2.5.1 Hydraulics of Overland Flow
Knowledge of the hydraulics of the surface water runoff is
needed for the detailed understanding of the general soil erosion
process. Many studies have been completely dedicated to this complex
overland flow hydraulics. Surface runoff is the most dynamic part of
the response of a watershed to rainfall.
The runoff from a watershed can be subdivided in sheet flow;
rills and gullies flow; and openchannel flow. Overland flow deals
basically with the first two kinds of flows and it is the one which
supplies water and sediment to the open channels. The equations
192
conditions were assumed. Overland flow with rainfall is very complex
and simplifying assumptions like uniform flow conditions may not be
the best representation to be used.
To the author's knowledge, the best data available in the
literature which could be used for this study are the experimental
results published by Kilinc and Richardson (1973). In their investi
gation they measured the amount of soil eroded from a 16 ft (4.88 m)
long by 5 ft (1.52 m) wide uniformly sloped flume when a constant
rainfall intensity was applied. Twentyfour runs were made over bare
sandy soil using six different slopes (5.7% to 40%) and four rainfall
intensities 1.25 to 4.60 in./hr (32 to 117 mm/hr). Tables 4.1
through 4.3 show the parameters and computations obtained from the
data. Details about each parameter are discussed in the following
sections.
4.2.1 Soil Properties
Kilinc and Richardson (1973) reported that all runs had cons
tant dry specific weight, y^, of 93.6 lb/ft3 (1.5 g/cm3) of soil
including pores and porosity of 43%. The specific weight of the
individual soil particles, was 164.2 lb/ft3 (2.63 g/cm3).
The soil can be classified by the Unified Soil Classification
System, ASTM D2847 (USAWES, 1967), as a silty sand soil. The grain
size distribution of their sandy soil showed that only 1.25% of the
original soil had a diameter larger than 2 mm and about 12% had a
diameter finer than 53 urn. The transported sediment was reported to
have 1% to 2% particles with a diameter larger than 2 mm and 5% to
204
Richardson because they used the critical value corresponding to the
grain size d50 of the transported sediment. No correction on
xcr due to the slope was made on that study.
4.2.6 TimeMean Bed Shear Stress Evaluation
The timemean bed shear stress x0 was obtained from Kilinc
and Richardson's data. Only the value of x0 obtained from the so
lution of the dynamic momentum equation (Equation 2.19) was used. It
is considered that the value of x0 is the most representative of
the existing conditions at the bed surface of the overland flow with
rainfall. The use of the DarcyWeisbach equation (Equation 2.28) or
Shen and Li's (1973) empirical equation (Equation 2.33) may not be
enough to account for all of the variations of x0 in overland flow
with rainfall.
4.2.7 Grain Reynolds Number Evaluation
The grain Reynolds number Rede was obtained from the
expression
Rede = v* de/v (4.5)
All of the information required to obtain Rede was ob
tained from Kilinc and Richardson's data. Values of Rede are
presented on Table 4.2.
4.2.8 Dimensionless Sediment Transport Parameters
4.2.8.1 Evaluation of $
The sediment transport intensity function, i>, was already de
fined by Equation 3.16. Einstein (1950), Chiu (1972) and other au
thors have presented their sediment load results in terms of this
dimensionless function. This study will also use that function for
319
As presented before, the regression analysis of Kilinc and
Richardson's data shows that the estimated value of the coefficient m
in Equation 4.13 was 0.78. The statistical analysis of the model
indicates that m can be equal to 1.0. Both cases are presented in
Figure 5.12 in the solution of Equation 4.13. This bed slope correc
tion factor for the saltation length does not depend on the water
depth or the rainfall intensity at the same time, like Equation 4.28
does. It only depends on the hydrodynamic forces acting on the grain
at the instant of detachment which also depends on the bed slope and
the corresponding incipient condition of the bed material.
After the grain is detached from the bed surface the concept
initially presented by Chiu for the saltation will be valid. How
ever, the initial hydrodynamic detachment force has a lower magnitude
which leads to smaller saltation length as the bed slope increases.
Figure 5.12 indicates that this slope correction factor approaches
zero as the bed slope angle approaches the angle of repose. This
does not mean that the erosion process will stop when both angles are
equal. This mean that all grains on the bed surface will move as a
single mass to a location where the bed surface has a smaller bed
slope angle and a new dynamic equilibrium may be established.
Also, at these high bed slope angles, saltation will approach
more to the rolling process (motion with full contact with the bed
surface) because only small hydrodynamic forces are needed to initi
ate that form of motion. However, the buoyant weight of the grain
will not allow the grain to separate too far from the bed surface.
32
Recently, Wright (1986) presented a physicallybased model of
the dispersion of splash droplets from a waterdrop impact on a slop
ing surface. He considered the forces and momentum transfer acting
at the moment of impact in order to obtain the velocity vectors of
the droplets. The absorption of some of the waterdrop's momentum by
the soil particles was considered as well as the air resistance ex
erted on the droplets while they travel in the air. The effects of
slope, wind, raindrop size and some soil properties on the droplet
distribution were also included. The probability of a particular
droplet size being transported was obtained from splash droplet size
distribution obtained from Mutchler studies. Although the proposed
model considered the soil absorption of the waterdrop momentum the
model does not consider the detachment of soil particles which would
be the next stage toward a model of soil erosion by rainsplash.
2.4.2 Splash Erosion Studies
The literature shows many studies dedicated to the splash ero
sion. There are studies about: techniques used to measure the
splash erosion, soil and rainfall properties which are important in
this process, mechanics of the process, rate of soil detachment with
respect to time or to rainfall intensity, empirical relationships to
represent the erosion rate of this process, etc. Not all of the pub
lished studies can be presented here but at least a brief description
of the current stage of this erosion process is presented.
The most popular method used to measure the splash erosion con
sists of exposing a small amount of soil in a cup to the direct
170
Using the complete momentum equation (Equation 2.19) the value
of the timemean bed shear stress (x0) can be obtained. But, the
real values that are necessary to obtain are the instantaneous bed
shear stress (t0) and the instantaneous lift force per unit area \
exerted on the grain at any instant. As mentioned in the saltation
length section, a grain will be detached when the combined effect of
lift and drag can overcome the buoyant weight of the grain. This
condition occurs when the resultant force of these forces makes an
angle with respect to the normal of the theoretical bed surface
greater than the angle of repose of the bed material.
The condition for the grain being about to be moved (also
called incipient motion) is presented in Figure 3.4 for the case of
horizontal bed and sloping bed. For the case of the sloping bed, the
buoyant weight is resolved into two components. The net effect is
that smaller magnitudes of the lift and drag forces than the ones
observed on the horizontal bed are enough to reach the incipient
motion. Usually, the longitudinal slope in rivers is very small
(less than 2%) but, for the case of overland flow, the longitudinal
slope can reach values of the order of 20% to 60% in some places.
Therefore, slope effects on the erosion criterion must also be consi
dered. Chiu's criterion was based on horizontal or nearly horizontal
beds. For this study the sloped bed case is presented.
As shown in Figure 3.4 erosion takes place on a longitudinal
sloped bed when
384
Kisi sel, I. T., R. Rao, J. W. Delleur, and L. D. Meyer, "Turbulence
Characteristics of Overland FlowThe Effects of Rainfall and
Boundary Roughness," Water Resources and Hydromechanics
Laboratory, Technical Report No. 28, Purdue University, West
Lafayette, Indiana, 1971.
Kneale, W. R., "Field Measurements of Rainfall DropSize Distribu
tion, and the Relationships between Rainfall Parameters and Soil
Movement by Rainsplash," Earth Surface Processes and Landforms,
7, 499512, 1982.
Knisel, W. G., "CREAMS: A FieldScale Model for Chemicals, Runoff
and Erosion from Agricultural Management Systems," U. S. Dept,
of Agriculture, Conservation Research Report No. 26, 1980.
Komura, S., "Hydraulics of Slope Erosion by Overland Flow," J. of
Hydraulics Division, ASCE, 102(Hy10), 15731586, 1976.
Kramer, L. A., and L. D. Meyer, "Small Amount of Surface Mulch Reduce
Soil Erosion and Runoff Velocity," Trans, of the American
Society of Agricultural Engineers, 12, 638641, 645, 1965.
Lane, L. J., and E. D. Shirley, "Modeling Erosion in Overland Flow,"
in Proc. of the Workshop on Estimating Erosion and Sediment
Yield on Rangelands, Tucson, Arizona, March 79, 1981, U. S.
Dept, of Agriculture, Agricultural Reviews and Manuals,
ARMW26, 120128, 1982
Lattanzi, A. R., L. D. Meyer, and M. F. Baumgardner, "Influences of
Mulch Rate and Slope Steepness on Interrill Erosion," Proc. of
Soil Science Society of America, 38, 946950, 1974.
Laufer, J., "Investigation of Turbulent Flow in a TwoDimensional
Channel," NACA Technical Note 2123, National Advisory Committee
for Aeronautics, 1950.
Laursen, E. M., "The Total Sediment Load of Streams," J. of the
Hydraulics Division, 84 (Hyl), 15301533, 1958.
Laws, J. 0., "Recent Studies in Raindrops and Erosion," Agricultural
Engineering, 21(11), 431433, 1940.
Laws, J. 0., "Measurements of Fal1Velocity of Water Drops and
Raindrops," Trans, of American Geophysical Union, EOS, 22,
709721, 1941:
Laws, J. 0., and D. A. Parsons, "The Relation of RaindropSize to
Intensity," Trans, of American Geophysical Union, EOS, 24,
452460, 1943*:
115
records of erosion collected on the eastern part of the United
States. Many parameters or factors which could possibly be related
to soil erosion were studied (Smith and Wischmeier, 1957; Wischmeier
and Smith, 1958; Wischmeier et al., 1958) and the ones more closely
correlated to the soil loss were selected.
The word 'universal' was suggested because the equation is free
of some of the generalizations, and geographic and climatic restric
tions inherent in earlier models. It has been criticized as not be
ing universal because the parameter values were presented for condi
tions of the eastern twothirds of the United States. However, as
more data are accumulated every year the USLE parameters have been
evaluated or identified in more areas, including regions on other
continents. Currently the USLE is used worldwide.
As indicated in the literature, this equation was developed to
estimate longterm average annual soil loss. Therefore, its applica
tion to a specific year or storm may not be appropriate. When used
with a specific storm it will estimate the average soil loss for
numerous recurrences of that event which may vary considerably with
the measured soil loss of that storm. In addition, some of the fac
tors used in the USLE are obtained from diagram and tables obtained
from specific experimental data. Consequently, the application of
this equation to situations for which the factor values are not yet
determined is dangerous and extrapolation is hazardous. Other limi
tations of the USLE is that it does not estimate deposition (Wisch
meier, 1976) and cannot estimate gully and channel erosion.
354
A statistical analysis was conducted to support the proposed
model and the assumptions made during the development of the equa
tions. The errors in the predicted values are very similar to the
estimated errors of the data used. The maximum relative error in the
predicted values was approximately 33%, which can be considered
reasonable for the complex process of soil erosion by overland flow
with rainfall.
An expression to predict the normalized velocity fluctuations,
su, was developed based on the particle Reynolds term v*de/v. It was
also found that the relationship between su and the sediment
concentration, Cs, was also satisfied by the same expression of
su and v+de/v. This expression appears to be independent of any
other rainfall or bed slope effect other than the ones already
included in the v* definition. This expression can only be used
for the flow conditions for which it was developed. The suvalue
is restricting the general use of the proposed model. If su is
known or any expression to obtain su is available, the proposed
model may be able to predict the rain and/or flowinduced transport
of cohesionless materials.
The redefinition of the saltation length of individual parti
cles presented here, recognizes the decrease of the required detach
ment forces as the bed slope increases. This may lead to a shorter
saltation length of the particle as the bed slope increases. The
complex interrelationship between the rainfall intensity, the bed
slope, and the water depth on the saltation length was also
CHAPTER II
SOIL EROSION PROCESS AND REVIEW OF RELATED STUDIES
2.1 The Soil Erosion Process
The initial cause of soil erosion due to water is rainfall.
When the raindrops impact a ground surface not covered by water,
their kinetic energy will generate a splash of water in which thou
sands of droplets will disperse in all directions (Mutchler and Lar
son, 1971; and Mutchler, 1971). Some of these droplets will carry
soil particles out of the area of impact. The amount of soil de
tached and the distance traveled by the individual soil particles
will be a function of the ground surface soil properties and the
rainfall characteristics.
If there is a water layer covering the ground surface, the ef
fect of the splash on the soil surface will be mostly a function of
the water layer thickness, the drop diameter and the soil properties
(Mutchler, 1967; Mutchler and Young, 1975). These authors found that
raindrop impacts are more erosive when the water depth is about one
fifth of the drop diameter and that the impacts are practically non
erosive when the soil is covered by water at a depth of about three
drop diameters or more.
Palmer (1963, 1965) also studied the effects of the impact of
waterdrops with the shallow water layer. He studied the stress
strain relationship on a surface covered by different water layer
thicknesses that were impacted by different drop sizes. His maximum
7
382
Hirschi, M. C., B. J. Barfield, and I. D. Moore, "Modeling Soil
Erosion with Emphasis on Steep Slopes and the Rilling Process,"
Water Resources Research Institute, Research Report No. 162,
University of Kentucky, Lexington, Kentucky, 1985.
Hjulstrom, F., "The Morphological Activity of Rivers as Illustrated
by River Fyris," Bull, of Geological Institute of Uppsala,
Sweden, 25, Ch. Ill, 1935.
Hobbs, P. V., and A. J. Kezweeny, "Splashing of a Water Drop,"
Science, 155, 11121114, 1967.
Hobbs, P. V., and T. Osheroff, "Splashing of Drops on Shallow
Liquids," Science, 158, 11841186, 1967.
Holy, M., Erosion and Environment, Pergamon Press Ltd., Oxford,
England, Translated by Jana Ondr^kov, 1980.
Horton, R. E., "The Interpretation and Application of Runoff Plate
Experiments with Reference to Soil Erosion Problems," Proc. of
Soil Science Society of America, 3, 340349, 1938.
Horton, R. E., "Erosional Development of Streams and Their Drainage
Basins; Hydrophysical Approach to Quantitative Morphology,"
Bull, of the Geological Society of America, 56, 275370, 1945.
Horton, R. E., "Statistical Distribution of Drop Sizes and the
Occurrence of Dominant Drop Sizes in Rain," Trans, of American
Geophysical Union, EOS, 29(5), 624630, 1948.
Huang, C., J. M. Bradford, and J. H. Cushman, "A Numerical Study of
Raindrop Impact Phenomena: The Rigid Case," J. of Soil Science
Society of America, 46, 1419, 1982.
Huang, C., J. M. Bradford, and J. H. Cushman, "A Numerical Study of
Raindrop Impact Phenomena: The Elastic Deformation Case," J. of
Soil Science Society of America, 47, 855861, 1983.
Hudson, N. W., "Raindrop Size Distribution in High Intensity Storms,"
Rhodesian J. of Agricultural Research, 1, 611, 1963.
Hudson, N. W., "The Influence of Rainfall on the Mechanics of Soil
Erosion with Particular Reference to Rhodesia," M. S. Thesis,
University of Cape Town, Republic of South Africa, 1965.
Hunt, R. E., Geotechnical Engineering Investigation Manual, McGraw
Hill Book Co., New York, 1984.
Izzard, C. F., "The SurfaceProfile of Overland Flow," Trans, of the
American Geophysical Union, EOS, 25, 959968, 1944.
I
Â§
e
=a*>
0.5
0.4
0.3
02
Q1
0
0.1
0.2
0.3
0.4
0.5
6 5 4 3 2 1 0 1 2
2<*err
0
o
1err
o
u
r LEG
END
Rainfall
0
^ ^err
inVhr mm/hr
o
L
1.25
2.25
3.65
4.60
32
57
93
117
j
o
3
^^OBS
Figure 5.9. Residual Values Versus the Natural Logarithm of Observed Â¡Values.
CO
o
133
Gregory and Steichen (1978) developed their soil erosion equa
tion from individual storm events based on physical principles in
volved in the detachment and transport of soil observed from experi
mental results by other researchers (i.e., Ekern, 1950; Meyer and
Monke, 1965; Palmer, 1965; and Foster and Martin, 1969). The erosion
equation could consider: variations in storm type (distribution of
rainfall intensity during the storm event), water depth to protect
soil from raindrop impacts, and changes in the flow conditions
through changes in the Manning's roughness coefficient. It could
also consider variations in the bulk density of the soil, infiltra
tion capacity, soil cover condition, and bed slope effects to gener
ate a peak soil loss as reported by Foster and Martin (1969). Com
pared to the USLE the proposed model appeared to better describe the
soil loss variation of single storm events due to variations in the
distribution of rainfall intensity within the storm duration.
Li (1979) developed an erosion model for watersheds using the
kinematic wave approach to obtain the hydraulic parameters and for
the sediment routing he divided it into suspended load transport and
bedload transport. Einstein's (1950) approach was used for the
suspended transport equations and the bedload was expressed similar
to what Einstein assumed
qs = 11.6 Ca v* ab (2.75)
where
Ca = sediment concentration near the top of the bed layer
ab = thickness of the bed layer which was considered as twice
the size of the sediment particles
1 2 4681 2 4681 2 4681
10 4 Cs (ppm)
10
6
Figure 4.9. Predicted suValue Versus Measured Sediment Concentration
using C* = 2.2 x 10^ ft~^ (Constant).
ro
C
287
soil characteristics, rainfall conditions and flow conditions are
within the limits of the data used for the model calibration.
The limits of the use of this model are rainfall intensities
between 1.25 in./hr (32 mm/hr) to 4.60 in./hr (117 mm/hr); bed slopes
between 5.7% and 40%; flow Reynolds number, based on mean velocity
and water depth, between 20 and 135; particle Reynolds numbers, based
on effective grainsize and friction velocity, between 2.75 and 7.4;
mean water depth from about two times the effective grainsize diame
ter to about seven times the effective grainsize; and effective
grainsize around 0.15 mm, i.e., corresponding to fine sand. These
conditions are the limits of Kilinc and Richardson's data. To expand
these limits, data beyond these ranges are necessary but this writer
was not able to find any other reference of overland flow with rain
fall with all the information required to use the model. Other con
siderations will be presented throughout this chapter.
In Table 5.9 the statistical Ftest value obtained from the
comparison between the variance of the regression and the variance of
the residual (error) shows that the model can be accepted with a cer
tainty greater than 99% (Brandt, 1976) because:
525.1 >> F (4, 19, 0.99) = 4.50
Table 5.9 also indicates that the mean values of the observed
and predicted Â£n $ values are basically the same, and that the cor
responding sample standard deviation are basically equal. These num
bers change in the fourth significant figure. All of these consider
ations indicate that the model replicates the observed values very
well.
277
10
2 
10
2
2 
10 '3 1
O
LEGEND
Rainfall
in ./hr
mrrVhr
1.25
32
o
2.25
57
3.65
93
>
4.60
117
10 20 30 40
Bed Slope, S (percent)
50
Figure 5.5. Observed iValues for Given Bed Slope.
Horizontal Bed
Rainfall
H W U Hv.
^Control Surface
Sloping Bed
FLOW
h 7 H
Ax
RMFn = pp,IVtB Ax
P = density of water
I = rainfall intensity
Vt= raindrop terminal velocity
p = rainfall momentum flux
1 correction factor
RMF. = normal component of
N rainfall momentun flux
B = width of channel
Ax = longitudinal increment
0 = bed slope angle
Figure 4.7. Slope Effects on the Normal Component of the Rainfall Momentum Flux.
r\>
co
o
3
significant variables or parameters and studied their behavior under
different conditions. Unfortunately, most of these studies were con
ducted under simulated steady or quasisteady conditions. This is
considering that the surface area under study is significantly larger
than the area affected by a single raindrop impact and that the time
period is substantially larger than the time increment between rain
drop impacts. Under such conditions, the soil erosion process can be
mathematically described as a simpler process, but the capability to
predict changes in time and sometimes spatial changes is lost.
There are other studies in which a specific part of the soil
erosion process is considered and measured. Then a mathematical
model is developed and used in describing that part of the erosion
process.
Some of the general topics related to soil erosion studied in
the past are
 Relation of rainfall to runoff and soil erosion.
 Detachment and/or transportation of soil particles by
raindrop impact.
 Detachment and/or transportation of soil particles by
runoff.
 Soil erosion caused by the combined actions of rainfall and
runoff.
 The erosion process related to hydraulic parameters.
 The influence of soil properties on the erosion process.
 The influence of longitudinal slope, slope length and/or
slope shape on erosion.
 Measurement and prediction of soil loss from a given area.
335
Hence,
Net change of E = 94.09
1 p
Relative error of .P = 94.,0? x 100 = 29.2%
1 p 321.58
Therefore, the conclusion is that the model has a mathematical
limitation when the probability of erosion is near unity because it
is not possible to measure suvalues with that great accuracy. A
certain percentage error must be accepted in the solution of the
model as the probability of erosion approaches unity. The accuracy
with which su and are measured have a certain physical minimum
value which depends on the techniques and procedures used. Using the
minimum value as a reference point, it can be deducted that the
percentage error of the solution will grow as the probability of
erosion approaches unity (i.e., for p = 0.985 or greater).
Figure 5.15 presents the changes in ydue to errors of p.
Curves with 2%, 1%, 0.2% and 0.1% absolute error are presented with
respect to the normal distribution probability curve (perfect agree
ment curve). As shown in this figure, an absolute error of 2% of p
may not significantly affect the prediction of $ by Equation 3.17 if
the probability of erosion is less than 80%. Similarly, it can be
summarized that the errors of $ will not be significant for
Absolute Error Probability of Erosion Range
1%
P
< 0.90
0.2%
P
< 0.98
0.1%
p < 0.99
28
Mutchler and Young (1975) presented a relationship for the rate
of change of width of the crater with time. From this, they obtained
an expression of the lateral (horizontal) velocity, u^ of the wa
ter moving away from the impact site along the surface. They also
obtained a rough estimate of the viscous bed shear stress t0 by us
ing the equation
where
du = rate of change of the horizontal velocity
dz = increment of vertical distance in the water
p = dynamic viscosity of water
Based on these conditions an estimate of the minimum velocity
required to detach soil particles from the surface and how long those
shear stresses could last before they become smaller than the criti
cal shear stress, tcr was presented.
From this approach Mutchler and Young were able to show that
the erosive action of a waterdrop impact was effective very early
after impact and thus in the vicinity of the center of impact. They
also showed that for water layer depths equivalent to threedrop di
ameters, the soil is essentially protected from raindrop impacts.
Finally, it was also indicated that most of the water splashed from
the area of impact came from the water layer and not from the water
drop itself.
III
3.65
VII
3.65
XI
3.65
XV
3.65
XIX
3.65
XXIII
3.65
IV
4.60
VIII
4.60
XII
4.60
XVI
4.60
XX
4.60
XXIV
4.60
3.035
12.74
2.467
12.89
2.052
13.02
1.819
13.06
1.578
13.29
1.449
13.35
3.238
16.30
2.701
16.56
2.248
16.83
2.052
16.96
1.801
17.00
1.617
17.02
5
10
15
20
30
40
5
10
15
20
30
40
Source: Kilinc and Richardson (1973)
0.2102
1.43
0.1697
1.39
0.1338
1.312
0.1238
1.24
0.0617
1.22
0.0446
1.222
0.2004
1.41
0.1299
1.39
0.0559
1.282
0.0208
1.282
0.0100
1.282
0.0046
1.282
2.4069
0.646
4.5837
3.72
5.1295
7.138
5.768
14.904
6.095
22.648
6.8225
37.00
3.1583
1.482
5.0945
5.88
6.1907
12.88
6.7906
26.66
6.722
37.52
7.6281
65.08
'>0
4^
Copyright 1987
by
Raul Emilio Zapata
Table 5.10 STATISTICAL ANALYSIS OF THE COEFFICIENTS IN THE MODEL
Used for
Evaluation
Coefficient of
Estimated
Value *
EV
Standard
Error of
Estimate
SEE
x 100
(percent)
Student's
tValue**
aj = 0.05
Check Null
Hypothesis
H0 = 0**
C5 ]
su
0.58
0.03558
6.1
16.301
Rejected
C* .
(Equation 4.41)
0.153
0.02036
13.3
6.494
Rejected
m
r
0.78
0.25878
33.2
3.014
Rejected
C2 '
(Equation 4.40)
2.3
0.30668
13.3
7.5
Rejected
C4 J
8.8
8.64025
98.2
1.018
Not rejected
Rounded off values. Some error can be reduced if more significant digits are used. However, the
unknown accuracy of data does not support the use of this approach.
**For 19 degrees of freedom and 5 % level of significance, the Student's tvalue is
t(IQ o.975) = 2*093 (Brandt, 1976).
213
Kilinc and Richardson's data were collected under steep slope
conditions. The effect of the bed slope can be observed on Figure
4.4 by selecting any curve with constant rainfall intensity and fol
lowing the corresponding data points. The general trend is that as
the bed slope increases the erosion increases ($ increases) and the
rate of erosion decreases (Ao decreases). The same general trend is
observed when the rainfall intensity is increased. This last general
trend is more obvious in Figure 4.5 where the data points are plotted
with same bed slope on the same curve. Each curve is practically a
straight line with a gradual steepness increase as the bed slope
increases.
These general trends will be discussed later. The proposed
physical model must account for them too. Some of the rainfall and
bed slope effects are already considered in the evaluation of the
timemean boundary shear stress, ^0, by the dynamic equation (Equa
tion 2.19) and on the correction of the timemean critical shear
stress due to the longitudinal slope, Tcrs The model considers
these two shear stresses related to each other by Equations 4.9 and
4.10. There are other rainfall intensity and bed slope effects which
need to be presented before the final evaluation and verification of
Equation 3.41. In the next section the evaluations for C+ and su are
presented. They include the rainfall intensity and bed slope effects
which may help to describe the erosion process on a physical basis.
139
During the last decade, the interest in water quality, nutrient
removal and pesticides control has also been reflected in soil ero
sion models.
ANSWERS, Areal Nonpoint _Source Watershed Environment Response
Simulation (Beasley et al., 1980), was developed in an effort to sup
ply information about the effect that land use, management, and con
servation practices or structures might have on the quality and quan
tity of water coming from primarily agricultural watersheds. Unlike
many large scale watershed model structures, this model used distri
buted (rather than lumped) parameters. It was developed on an event
(rather than long term) basis, and allowed determination of both spa
tially and temporal information. The input data was designed to
utilize information readily available from existing sources (e.g.,
SCS Soil Surveys, USGS topographic maps, crop and management sur
veys).
The erosion component of this model used the sediment continu
ity equation, Equation 2.69 for quasisteady conditions, as original
ly described by Foster and Meyer (1972b). The detachment rate due to
rainfall and overland flow followed the general shape of the equa
tions used by Meyer and Wischmeier (1969) but the USLE factors of
Cm and Kf (crop and management factor and soil erodibility fac
tor) were also included in the equations. The particle transport by
overland flow was expressed by two equations obtained from Yalin's
sediment transport equation, as follows
303
10"3 10'2 10*1 10 101
$ObB
Figure 5.8. Predicted $ Versus Observed ^Values
89
on the flow Reynolds numbers. In other words, the relative increase
in the magnitude of au with increasing Ref did not appear to com
pensate for the increase in the corresponding point mean velocity,
thereby resulting in a decrease in the relative turbulence intensity
( au/u).
The results presented by Kisisel (1971) relating to overland
flow with rainfall showed that close to the smooth boundary o /Â¡] did
not exhibit differences compared to the overland flow with no rainfall
values. However, it was reported that these relative intensities
increased substantially (about 100%) as the free surface was
approached.
For the case of flow over the rough boundary with and without
rainfall the o^/u values increased about 10% to 20% as compared to
the respective values in flows over the smooth boundary.
Both Yoon and Kisisel, indicated that for flows over smooth
boundaries with no rainfall thea^/G values were found to be dependent
on the distance from the bottom boundary and the Reynolds number. For
flow Reynolds numbers greater than 1500, both studies indicated that
close to bottom boundary y/h =0.1 the value of au/u was around 0.15
and as the water surface was approached the u/u values decreased
to about 0.04. Kisisel also reported that much closer to the bottom
the u/ti values were scattered and near the boundary showed values
between 0.3 and 0.2. Extrapolation of Yoon results indicate a o /G
value around 0.2 at the bottom. These values generally agree with re
sults obtained by other researchers who have studied boundary induced
291
the estimate of Cg indicates that the initial water depth can be
between the grainsizes d0O = 290 pm = 9.51 x 104 ft to <\c^ = 377 pm
= 1.24 x 10" 3 ft (50% and 57% of finer material, respectively).
The m value of 0.78 has a ratio of of 33.2%. This esti
mated relative error seems a little high. However, the null hypothe
sis is used to verify if the saltation length is not influenced by
the slope effects represented by Equation 4.13. The null hypothesis
was rejected at the 5% level of significance. This means that the
data indicate dependence of the saltation length on the longitudinal
bed slope effects represented by Equation 4.13.
It was mentioned in Chapter IV that the coefficient m can be
equal to one if the tangential component of the buoyant weight is
only implicitly included as an additional force able to increase the
saltation length.
Test for m = 1
0.78 1.0
0.259
= 0.85 < t
(19,0.975)
2.09
This hypothesis cannot be rejected for the 5% level of significance.
Therefore, it would be necessary to obtain additional data to statis
tically prove that m cannot be equal to one.
The value of C4 = 8.8 has a relative error of the estimated
value of 98.2%. This high relative error indicates that the true
value of C4 can be between C4 = 0.2 and C4 = 17.4. The test
for the null hypothesis also indicates that with a 5% level of
12
there were few studies in which the mechanism of soil erosion and
their effects were considered too.
Laws (1940) presented one of the first studies in which the
relation of raindrop size to erosion and infiltration rates were
considered. He also mentioned previous studies done by European and
American scientists around the turn of the century, and referred to
studies carried out at that time by himself and other researchers.
Ellison (1944, 1945, 1947, 1950) contributed a series of papers
in which he described the soil erosion process. It was the first
time this process was described and studied in such detail. Ellison
(1944) initially presented the current knowledge about the soil ero
sion process and the factors which might affect the process. He
developed an empirical equation for raindrop erosion (splash erosion)
based on the rainfall intensity, the diameter of the waterdrop and
the velocity of the drop. In 1945, he presented his experimental
results of the effects of raindrop impact and flow in the infiltra
tion capacity and the soil erosion. He divided the stuc(y in raindrop
effects alone, runoff effects alone, and the combined effects. Like
previous studies, many of his experiments were exploratory in nature
and the data had only qualitative significance.
Then Ellison (1947) proceeded to describe his approach to the
soil erosion problem step by step. He postulated that the soil ero
sion process was "a process of detachment and transportation of soil
materials by soil agents." This definition described the process as
To Carmencita,
Raul Enrique and Mari Luz
389
Mutchler, C. K., and C. L. Larson, "Splash Amounts from Waterdrop
Impact on a Smooth Surface," Water Resources Research, 7(1),
195200, 1971.
Mutchler, C. K., and R. A. Young, "Soil Detachment by Raindrops," in
Proc. of the SedimentYield Workshop, USDA Sedimentation
Laboratory, Oxford, Mississippi, Nov. 2830, 1972, U. S. Dept,
of Agriculture, Agricultural Research Service ARSS40, 113117,
1975.
Muzik, I., "Laboratory Experiments with Surface Runoff," J. of the
Hydraulics Division, ASCE, 100(Hy4), 501515, 1974.
Naden, P., "An Erosion Criterion for GravelBed Rivers," Earth
Surface Processes and Landforms, 12, 8393, 1987.
Nail, F. M., "Flume Studies of Sediment Transportation in Shallow
Flow with Simulated Rainfall," Water Resources Institute, Texas
A and M University, 1966.
Neal, J. H., "The Effect of Degree of Slope and Rainfall Characteris
tics of Runoff and Soil Erosion," Missouri Agricultural Experi
mental Station Research Bulletin, No. 280, Columbia, Missouri,
1938.
Nearing, M. A., and J. M. Bradford, "Single Waterdrop Splash
Detachment and Mechanical Properties of Soils," J. of Soil
Science Society of America, 49, 547552, 1985.
Nearing, M. A., and J. M. Bradford, "Relationship Between Waterdrop
Properties and Forces of Impact,", J. of Soil Science Society of
America, 51, 425430, 1987.
Nearing, M. A., J. M. Bradford, and R. D. Holtz, "Measurement of
Force vs. Time Relations for Waterdrop Impact," J. of Soil
Science Society of America, 50, 15321536, 1986.
Neibling, W. H. and G. R. Foster, "Sediment Transport Capacity of
Overland Flow," in CREAMS: A FieldScale Model for Chemicals,
Runoff, and Erosion from Agricultural Management Systems, OSDA
Conservation Research Report No. 26, Vol. 3, Ch. 10, 1980.
Olivier, H., "Through and Overflow Rockfill DamsNew Design
Techniques," Institution of Civil Engineers, Proc., London, 36,
433471, 1967.
Olson, R. M., Essentials of Engineering Fluid Mechanics, 3rd Edition,
Educational Publishers, New York, 1973.
93
the rainfall caused directional reversals while their magnitude in
creased especially toward the free surface points. The magnitude of
to was also reported to increase as a result of the rainfall but the
increase was mostly associated with the increase in water depth and
not as a function of the rainfall. This last statement may be due to
the flow conditions at which the study was performed. The flow was
alreacty in the turbulent range where the rainfall effects are not
predominant (i.e. Ref > 1900 for that study) as previously observed
in Yoon's (1970) results.
2.5.2 Overland Flow Erosion Studies
Runoff erosion can be subdivided into the three surface flow
patterns (i.e., sheet (or interrill), rill and gully flow) in which
water can transport sediments toward a permanent water body such as a
river or lake. How the very shallow water of the sheet flows is
responsible for the transportation of soil particles toward the more
concentrated and deeper flows in rills and gullies is a process which
many researchers have studied and tried to describe physically. Ini
tial studies, such as those alreacjy discussed in Section 2.2 of this
chapter, presented some of the basic ideas of the overland flow ero
sion process.
In addition to these studies, Horton (1945) presented a compre
hensive study of the erosional development of streams and their drain
age basins. In it, he suggested that the overland flow erosion did
not originate at the very top end of the sloped field where a no
erosion zone was initially required before the overland flow could
268
equation to use for the longitudinal slope correction factor. If
heavier bed materials are used, Equation 3.22 will approach Equation
3.21 which is also considered to be better at the static equilibrium
condition and for flow with to slightly higher than Tcr (e.g., Fer
nandez Luque and van Beek, 1976). Until further studies present more
evidence in support of Equation 3.22 or possibly Equation 3.23 the
use of the widely recognized Equation 3.21 appears to be satisfactory
for most of the cases and may be used without generating a signifi
cant error in the sediment transport formulas. Appendix B presents
the solution of the proposed model based on Equation 3.22 instead of
Equation 3.21.
5.2.7 Discussion of the Estimated Relative Errors of $and Tg
Table 5.4 shows the estimated relative errors of $, w1, and y '
e
using data from Kilinc and Richardson (1973). The results indicate
that the relative error off' is dependent on the bed slope. As the
bed slope increases, the relative error of 4^ also increases. This
is a consequence of increasing bed slope giving a lower threshold of
erosion (i. e., lower Tcrs). ^ also becomes more dependent on the
bed slope. However, it must be recognized that there is no random
ness on the relative error of v' presented here. Calculated relative
errors obtained from real data may show random errors around the
values shown here which assumed constant relative errors for each
parameter.
From Table 5.4, it is seen that the average value of the
estimated relative error of ^ is 26.8%, but with a strong dependency
on the bed slope angle.
221
hj = ZH cose
or, since ZH = hH
hj = hH cose (4.14)
Since this initial depth is a function of the effective grain
size it may be convenient to normalize Equation 4.14 by dividing it
by the effective grain size, de, and obtain
hj/de = (hH/de)cose (4.15)
Based on these considerations the effect of depth on the salta
tion length f(h/de) is approximated by using a Taylor series of the
dimensionless depth. To keep the expression as simple as possible,
all terms of second or higher order are neglected. The deep flow
with no rainfall is taken as the reference condition. Since the sal
tation length decreases as water depth decreases the depth effects on
the saltation length will be assumed to be represented by
f(h/de) = C0
Co
(h/de) C
1
C2 cosS '
(h/de)
(4.16)
where
f(h/de) = function to represent the water depth effects on
the saltation length
CQ = constant representing the saltation length on deep
water conditions
C2 = h^/de = constant representing dimensionless initial water
depth required for incipient grain motion on a
horizontal bed
h/de =
dimensionless water depth
339
For the flow conditions of this study, the probability of
erosion may be made to approach unity by increasing the bed slope or
by increasing the rainfall intensity. First, assume a constant
rainfall intensity while the influence of the bed slope increase is
studied. As the bed's inclination with horizontal approaches the
angle of repose the ^value decreases because the critical shear
stress (Tcrs) decreases and also because the bed shear stress (T0)
increases due to the increase of the bed slope. As 4^ decreases the
value of n+, the lower limit of the integral used to evaluate the
probability of erosion (Equation 3.40), becomes more negative. Con
sequently, the probability of erosion approaches unity.
Secondly assume the bed slope constant while rainfall intensity
is increased. In this case the frequency of the raindrop impacts and
the increase of flow discharge can increase the T0 value while the
tcrs remains constant. So the value will decrease as the rainfall
intensity increases. Consequently, the probability of erosion will
increase and approach unity as rainfall intensity becomes larger or
the flow discharge increases.
The combined effects of steep bed slopes and rainfall will, of
course, make p approach unity sooner. In this study, all data with
40% bed slope indicate probabilities of erosion in excess of 0.992.
There are also three points with bed slope of 30% and one point with
bed slope of 20% which have p equal to 0.985 or higher. All of these
points are susceptible to the above mentioned problem of su predic
tions and the corresponding errors in p and
CHAPTER VI
CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
The principal goal of this study was to develop a soil erosion
model for overland flow with rainfall based on physical concepts and
observations. The basic sediment transport equation used by the pro
posed model is based on the modified Einstein equation for total load
transport of noncohesive materials in open channels as presented by
Chiu (1972). The proposed model was developed as generally as possi
ble in order to be valid for the case of deep water flows (i.e.
rivers and open channels) as well as very shallow flows (overland
flow) with or without rainfall.
The concepts introduced to the development of the soil erosion
equation are
(1) Chiu's approach in expressing the saltation length of a
particle as inversely proportional to the particle diameter is used
here. However, the saltation length will also be affected by the
longitudinal bed slope, the water depth and the rainfall. With these
additional parameters the saltation length is more directly related
to the flow conditions. Chiu considered the saltation length to be
independent of the flow conditions.
(2) The rainfall effects on the erosion process are mostly
represented in the changes of the boundary shear stress and in the
352
169
determined in the next chapter. It should be recalled that the
expected solution will also be valid for open channel flows. This
means that the value obtained by Chiu of C* = 2.2 x 107 ft2 must be
used as a boundary condition to the new function for C* corresponding
to no rainfall condition in open channel flow.
3.3 Relationship Between Probability of Erosion and
Bed Shear Stress
3.3.1 The Erosion Criterion
Usually, the erosion in open channels is based on the bed shear
stress induced by the flowing water. For the case of overland flow
without rainfall the approach is practically the same as a deep open
channel because the flowing water is the only energy available to
generate erosion. When rainfall is also included as an eroding agent
the erosion process becomes very complex. As it was mentioned in the
literature review, many researchers have tried to fully explain the
erosion process, however only with partial success. This study
attempts to explain the erosion process by overland flow with rain
fall as general as possible without losing the simplicity.
Many researchers (i.e., Keulegan, 1944; Woo and Brater, 1962;
Yoon and Wenzel, 1971; Shen and Li, 1973; Woolhiser, 1975, and
others) have studied and recognized that rainfall increases the flow
resistance of the overland flow which also means an increase on the
bed shear stress. By increasing the bed shear stress, the probabil
ity of erosion increases and the amount of material removed from the
area under study also increases.
177
greater values of the longitudinal slope in the overland flow case, a
more detailed discussion of it mey be in place. Both Equation 3.21
and 3.22 can be derived from the equilibrium of forces or the moment
of forces acting on a particle of the surface layer exposed to the
flowing water. Equation 3.21 predicts that no flow is necessary to
reach incipient motion when the bed slope angle is equal to the angle
of repose, i.e., the static (no flow condition) is accounted for by
this equation.
Equations 3.21 and 3.22 may lead to a more general concept.
The buoyant force is always normal to the water surface and the
hydrostatic pressure at any location in the water is given by
Yh cose. Based on this, the general equation for the correction
factor due to longitudinal slope may be expressed as
S1pe = 5 (cose sine cot) 
Y Y
0 1 Hor.
Y Y
S
cose [cos(ee) sin(eÂ£)cot] (3.23)
where
= water surface angle with respect to the horizontal
When the water surface is horizontal, i.e., Â£ = 0, Equation
3.23 gives Equation 3.21 and the noflow condition for incipient mo
tion can be satisfied. Similarly the uniform flow condition proposed
by Ulrich is obtained when Â£ = 6 is used in Equation 3.23. The
Table 2.1 RANGE OF VALUES OF VARIABLES AND SOME DIMENSIONLESS PARAMETERS IN OVERLAND FLOW
Model
Prototype
Variable
Units
Range
Typical
Range
Typical
Velocity, U
m
ft/sec
0.05 to 1
0.2
0.1 to 4
0.05
Depth, h
ft
0.005 to 0.05
0.02
0.02
Overland flow
ft
1 to 50
1.5
200 to 3000
1000
length, Le
Flow Reynolds number,

25 to 1000
100
100 to 20,000
800
Ref = Â¡Â¡mh/v*
 2 2*
1 +3 **
1
2 3
1
gh/um = 1/Fr

0(10 ) to 0(10 )
0(10 )
0(10 ) to 0(10 )
0(10 )
B


o
o
H
o

o
o
H
o
Friction coefficient, Cf
_
0(102) to 0(10_1)
__
0(10"3) to 0(102)
_
h/Le

0(104) to 0U0'1)
0(102)
0(106) to 0(10"3)
o
*
o
1
cn
I/m

o(io5) to odo"1)

0(106) to 0(10"3)

V^m

0(10) to 0(103)

0(10) to 0(103)


0(106) to 0(10~3)

Fr = Froude Number and v = kinematic viscosity of water.
** (10n) = Order of magnitude of 10n.
'ir'ir 4"
No infiltration in the model.
Source: Grace and Eagleson (1965, 1966).
Table 4.3 CRITICAL SHEAR STRESS AND DIMENSIONLESS SEDIMENT TRANSPORT VARIABLES
Run
Shields
Entrain
ment
Function
Eh
Critical
Horizontal
Bed, V
(lb/ft2)
xlO3
Shear Stress
Sloped Bed
T
crs
(lb/ft2)
xlO3
Flow Intensity
Horizontal
Function
Sloped
V
Sediment
Transport
Intensity
Function
4>
I
0.0741
3.59
3.230
4.515
4.063
0.00777
V
0.0786
3.81
3.134
2.179
1.793
0.02382
IX
0.0741
3.59
2.628
1.924
1.409
0.04439
XIII
0.0716
3.47
2.224
1.431
2.917
0.05215
XVII
0.0716
3.47
1.597
1.179
0.543
0.07466
XXI
0.0716
3.47
0.990
0.986
0.281
0.09183
II
0.0799
3.87
3.482
2.623
2.361
0.02431
VI
0.0787
3.81
3.134
1.263
1.039
0.12219
X
0.0799
3.87
2.833
1.080
0.791
0.24103
XIV
0.0787
3.81
2.442
0.960
0.615
0.46074
XVIII
0.0741
3.59
1.652
0.902
0.415
0.82216
XXII
0.0741
3.59
1.024
0.800
0.228
1.0608
121
The literature presents many modified USLE equations based on
additional considerations which the original USLE did not have. Some
of the additional considerations include the works done by Onstad et
al. (1967) and Foster and Wischmeier (1974) in which the application
of USLE was adapted to irregular slopes and soil loss estimates from
individual storm basis. Then, Onstad and Foster (1975) used these
ideas and redefined the rainfall erosivity factor, R, as a function
of both rainfall and runoff energy expressed as
R = 0.5 Rei + 15 VR(qp)1/3 (2.62)
where
Rei = Rainfall factor as originally presented in the USLE
VR = storm runoff volume expressed in inches
qp = storm runoff peak expressed in inches per hour
The coefficient values were obtained based on the assumption that
detachment by rainfall and by runoff was about evenly distributed.
Deposition was also considered by comparing the sediment load from
the previous section plus the detachment at the section versus the
transport capacity at the lower end of the section. The relative
contribution of rills and intern'll erosion were presented based on
works of Young and Wiersma (1973) and Meyer et al. (1975b) which
showed that interrill erosion rates rise rapidly with downslope
distance and level off to a constant value while rill erosion rates
rise nearly linearly throughout the length of a sloped bed.
301
the curves of Figures 4.5, 5.2 and 5.4 until the solution of Equation
4.37 is obtained. The corresponding suvalue of data point VI
shows an absolute error in su of about 0.025 which might be very
significant. However, su will be fully discussed later (Section
5.5) in this chapter.
Data point XII was already mentioned as one of the potential
data points with large errors in Yg. As shown in Figure 4.5 ( ver
sus Vq for given slope) the point corresponds to the 15% bed slope
curve. This curve does not follow the general straight line pattern
of the rest of the curves. This comes from the presence on the curve
of more than one data point with significant errors of i> and/or y'.
By observing the pattern of the other curves in the figure and fol
lowing the behavior of the data points with rainfall intensity of
4.60 in./hr (117 mm/hr), it is possible to conclude that the observed
value of v' for point XII was lower than it should be. A higher val
ue of Yg, resulting from a decrease of T0, would improve the
straight line relationship shown in Figure 4.5. It would also im
prove the prediction shown in Figure 5.6 because the model's predict
ing curve is on that side of the observed value. The relative
0
errors of $ and Cs for data point XII are almost the smallest ones
of all of the data. In addition, Figure 5.5 does not show any irreg
ularity of $ around that data point. Therefore, the error of data
point XII must be coming from errors in the and y' evaluation.
t0 errors will affect the prediction of the suvalue because the
304
by a normal distribution it can be assumed that the error of the pre
sent model is also normally distributed.
Table 5.13 presents the predicted $values with the respective
absolute error and relative error of each point. It also presents
the standardized residual of the logarithmic values of $ which is
expressed as:
Standardized = Â£n ^OBSERVED ~ in ^PREDICTED (5 13)
Residual Standard error of the estimate
The standardized residual shows how far apart the predicted 4>
values and the observed ^values are based on the magnitude of the
standard error of the estimate. Special attention should be paid to
points with a deviation of more than two times the standard error.
If there is no logical explanation for that error, the model should
also be reevaluated in order to verify its validity at the region
where the large error in the data points is located.
Fortunately, the proposed model does not have any point with an
error greater than 1.56 times the standard error of the estimate.
However, a study was conducted for all data points in which the model
indicated an error greater than the standard error of the estimate.
The reasons for the errors in most of these points have been pre
sented previously in this chapter. Table 5.14 presents a summary of
these reasons for the possible error, and the figures and tables of
this study which support the explanation.
223
The rainfall effects and conditions on the saltation length are
too complex to fully account all of them in a single parameter. Some
simplifying assumptions are required. There are at least three dif
ferent raindrop impact conditions to be considered: deep water im
pacts, direct (no water layer) impacts, and shallow water impacts.
If the water depth is sufficiently deep, the effect of raindrop
impact will be completely absorbed by the water. This will totally
eliminate the rainfall detachment and overland flow will be the only
source capable of detaching grains from the soil surface. The salta
tion length can be considered as in an open channel with no rain
fall.
Direct raindrop impacts can detach soil particles into the air
sending them to longer distances than the ones they would travel
through water in a single jump. This case can basically occur at the
upstream end of the area under study for any rainfall conditions. It
can also occur at any part of the area if the rainfall intensity is
so low that it does not generate surface runoff. Usually this condi
tion can be neglected from studies in which the area not covered by
water is very small compared to the total area under study. In the
present study, it is assumed that a water layer covers the total
area, i.e., the direct raindrop impact condition is neglected.
This study is especially considering the case of shallow water
raindrop impacts. Here, the raindrop strikes the water layer first
and loses some of its energy or capacity to detach soil particles.
Therefore, the amount of soil material being detached by the raindrop
103
represented the prime sediment source in the gullies. The debris and
loose soil were found to mostly accumulate in the gullies during the
no rainfall season (winter) and then about 80% of them was eroded
during the spring rainfall season. The remaining 20% material was
washed away during other rainfalls along the year. Due to the high
rate of flow in the gully, most of the material carried into the gul
ly by the overland flow and rills can be also transported through the
gully.
Blong et al. (1982) studied the erosion of the sidewalls in
gullies. From field observations they found that the overland flow
and throughflow on a single side of the gully were responsible of 10%
to 30% of the gully volume. In the cases in which a vertical cut was
made in order to create a gully condition, the material lost by side
wall erosion was found to be responsible for more than half the gully
volume.
Piest et al. (1975b) concluded that gully erosion and its flow
transport rate were capriciously related to hydrologic and site loca
tion variables. But more overall knowledge about the gully erosion
mechanics is needed in order to use better conservation and protec
tion practices to control gully erosion.
2.7 Soil Erosion Estimates and Prediction
The literature presents many soil erosion models from empirical
and analytical (deterministic or stochastic) approaches. Some of
them are characterized by simplicity in their application while
others are more complex because they intend to account for many of
the different conditions "resent in overland flow with rainfall
328
slope of the straight line is now negative (i.e., su decreased as
v*de/v increased). Kilinc and Richardson did not present any tur
bulence information in their study because that was not the purpose
of their study. Therefore, no direct explanation can be obtained
from the original study. However, a physical explanation will be
attempted here.
For deep water conditions with no rainfall the variation of
su depends on the bed roughness, the grain shape and arrangement,
and the bed forms. Some researchers, such as Blinco and Partheniades
(1971), have also indicated a dependence of the longitudinal velocity
fluctuations on the flow Reynolds number. For shallow flow condi
tions with rainfall the same parameters can affect su but other
parameters, such as rainfall intensity, water depth, and relative
roughness will also influence the suvalue.
For overland flow conditions without rainfall, the flow regime
is basically viscous. When rainfall is included, the raindrop im
pacts induce turbulence to the flow. However, the agitation and
velocity fluctuations will die out as soon as the rainfall comes to
an end. The raindrop impact mcy have no significant effects on the
bed surface if the water layer is thick enough to absorb the impact.
Therefore, the suvalues (velocity fluctuations at the top of the
grains) in deep water mcy not be significantly different. There will
be an increase in the velocity fluctuations close to the water sur
face but the effects will gradually diminish toward the bed surface.
ro
Figure 2.5. Surfaces of Detachment, Transport and Maximum Erosion Rate (after **
Rowllson and Martin, 1971; reprint with permission of the American Society of Civil
Engineers).
Horizontal Bed Surface nnuu Sloping Bed Surface
L Lift Force
D = Drag Force
W = Buoyant Weight
X = Instantaneous lift per unit area
x0= Instantaneous bed shear stress
f = Resultant Force
rR
<) = Angle of repose
0 = Angle of longitudinal Slope
Figure 3.4. Forces Acting on a Grain About to Move for the Horizontal Bed and Sloping Bed
Conditions.
182
Using Shields' definition of the timemean critical bed shear stress
on horizontal beds
V = Eh<^ y>e <244
the timemean critical bed shear stress for sloped beds may be
obtained from Equation 3.21 and expressed as
\r = ^crs = Eh^Ys Y^de ^cose sine cot
sloped
Introducing this expression into Inequality 3.24 the instanta
neous bed shear stress required to reach incipient motion becomes
IS (3.26)
A1 Eh (x/ I To I ) + cot *
Many researchers have considered the stochastic nature of tq
(e.g., Chiu, 1972; Christensen, 1975, and others) by relating it to
the velocity fluctuations near the top of the grains. Following
their approach, the instantaneous bed shear stress may also be
written as
IT0 I \ CT 0 ut2 = ICT =t2(1 + Ut/St)2 (3'27)
where
ut = t + ut = 1nstantaneous velocity near the top of the grains
Ut = timemean velocity near the top of the grains
u^ = velocity fluctuations near the top of the grains
Using Equation 3.27 the timemean bed shear stress can be
expressed as
175
Equation 3.21 but the scarce data available in the literature do not
quite appear to verify Equation 3.22. It appears that Equation 3.21
better approximates the experimental values.
Ulrich has indicated that one possible reason for the discre
pancy between Equation 3.22 and the observed values is the definition
of the angle <)>. He indicated that the average angle of repose of the
bed material should not be used and instead he proposed the use of
the bearing angle between adjacent particles. For the coarse materi
als which he studied, the bearing angle was suggested to be around
75. This angle is based on a very short bed slope length. However,
the results presented by Ulrich in Figure 7 of his study, show that
the average bearing angle of the grains may decrease as the slope
length increases. Extrapolation of these results may indicate that
the average value may be approximated by the angle of repose of the
bed material when the slope length is around 3G times the mean grain
size diameter of the bed material or more. Consequently, for practi
cal purposes and contrary to Ulrich indications, the angle of repose
may be used as the definition of the angle <1. He recognized that the
present stage of knowledge is limited and further studies are needed
before a full explanation can be offered. As a comparison, Figure
3.6 presents the curves obtained from Equation 3.21 using = 40 and
Equation 3.22 using = 40 and = 75 as suggested by Ulrich (1987)
for his coarse material.
The variation of the pressure gradients direction may be more
significant in the case of overland flow than for rivers. Due to the
350
The Yalin (1963, 1977) equation assumes that the water depth is at
least ten times the grainsize diameter or higher. The Yalin bedload
equation is obtained based on this assumption. The present study
proposes an equation which allows consideration of the water depth
effects on the saltation length, for water depth of about two to
three times the effective grainsize diameter for overland flow
without rainfall. In addition, the proposed equation may reduce the
minimum water depth if the bed slope is too steep or rainfall is
present.
The data reported by Kilinc and Richardson has been used to
develop other soil erosion models. Li et al. (1977), used this data
in the development of a timedependent erosion model and reported
that their model could simulate the timedependent erosion rates to
the order of 30% for the tested data. Their comparison between the
computed and measured mean erosion rates showed good agreement at
most of the points but failed to agree at most of the points of the
1.25 in./hr (32 mm/hr) rainfall intensity.
The model proposed here has not directly included the time
dependence but the error of the mean erosion rates are of the order
of 30% also. This proposed model might show the time dependence if
fully timedependent data are tested. The same consideration can be
made if spacedependent data along the bed slope are tested. To use
the proposed model under these conditions, the previously mentioned
hydrologic model will be needed in order to obtain the required data
for this model.
UJ
p*
Figure 5.16. Required suValue Versus Measured Sediment Concentration, Cs
59
(1970) and in Yen (1972) for overland flow with or without rainfall
over smooth boundary case. Then each loss gradient term was written
in DarcyWeisbach resistance coefficient form (Equation 2.28) and
showed that each slope term was numerically different to each other.
The difference in magnitude among the coefficient depended on the
flow conditions (i.e., with or without rainfall or lateral rainfall).
Based on their results, they suggested that for steady spatially var
ied flow computations the momentum equation was preferred to the
energy equation or total head equation, particularly if the Darcy
Weisbach's Cf, Manning's NM, or Chezy's C coefficients are used
as the resistance factors.
Shen and Li (1973) also developed equations for the friction
factor and other parameters for rainfall conditions over "smooth"
surfaces based on the ratios of each parameter value (i.e., water
depth, water discharge, mean velocity, boundary shear stress, Froude
number and friction factor) under rainfall conditions and the equiva
lent parameter without rainfall with the same flow discharge rate.
This is like using the flow Reynolds number (Ref) as the scaling
number. They also obtained Equation 2.33 to calculate the friction
factor in the laminar range (Re < 900). For the very turbulent
regime (Re > 2000), the friction factor was obtained from Blasius'
equation but with a different coefficient value for rainfall condi
tions. For the intermediate flow regime (900 < Ref < 2000) a lin
early interpolated equation was proposed. These previous equations
were obtained by regression analysis of their data and from the lit
erature.
64
longitudinal distance from the upper end of the slope. This rela
tionship had a certain limit which was associated with the change in
the flow regime from laminar to turbulent flow. The onethird power
was also associated with Equation 2.25 used in the kinematic wave
method given that the flow discharge per unit width is expressed as
q = lx.
Yoon (1970) presented in his doctoral dissertation very signi
ficant information about xo over physically smooth surfaces when sim
ulated rainfall was applied to overland flow. His measured values
obtained from a flat surface hotfilm sensor agreed very well with
the computed T0 values from the one dimensional spatially varied flow
equation developed by Yen and Wenzel (1970), Equation 2.20a. This
showed the applicability of the onedimensional dynamic equation of
spatially varied flow for practical purposes.
Yoon indicated that, for a constant flow Reynolds number, tq
increased appreciably with increasing rainfall intensities. This
happened for Refvalues of up to approximately 1000. The rainfall
intensity effect became negligible as Ref further increased. He
also showed that the relationship expressed by Equation 2.30 was val
id for overland flow with rainfall and Ref < 1000 with the constant
K increasing with an increasing rainfall intensity and with a small
dependence of the bed slope. Later, Shen and Li (1973) indicated
that the slope effect was not significant on the Kvalue. So both
studies clearly showed that the flow Reynolds number and rainfall
intensity are the most important parameters affecting the timemean
boundary shear stress.
345
There are not enough data to allow development of these func
tions, but it is suggested that future research move in that direc
tion. Without knowing the suvalues the use of the proposed model
is limited to the experimental conditions. As mentioned at the end
of Chapter III, the proposed model can be used for rivers and any
other deep flow condition because the solution of this model ap
proaches Chiu's sediment transport model as the water depth increases
and the bed slope becomes nearly horizontal. This is based on known
suvalues under those conditions. If the suvalues are not
known, the proposed Equation 4.41 for predicting su may not be used
until it is validated at those flow conditions or another relation
ship is developed instead of Equation 4.41.
5.6 Final Remarks
After studying in greater detail the assumptions made for the
proposed model and the conditions in which Kilinc and Richardson col
lected the data, the following observations are presented.
The model has been prepared for practically any flow condition
and not only for overland flow with rainfall. But the evaluation of
the coefficients has been restricted to that very narrow flow condi
tion at the lower limiting boundary of the practically infinite range
for which it was developed. As mentioned previously the upper boun
dary corresponds to deep water flows where the proposed model ap
proach Chiu's sediment transport model.
The general use of this sediment transport model depends on the
evaluation of certain hydraulic parameters, such as the flow
19
relationship between fall velocity and fall height was also obtained
using the balance of forces equation in an integral form.
Beard (1976) studied the waterdrop behavior in the atmosphere
dividing the analysis in three physically distinct flow regimes
1 pm < De < 20 pm with 106 < ReA < 0.01
20 pm < De < 1 mm with 0.01 < ReA < 300
1 mm < De < 7 mm with 300 < ReA < 4000
where
De
%
= equivalent spherical drop diameter
Vn Dp
= ~ = drop Reynolds number
Vq = drop velocity
VA = kinematic viscosity of air
For each regime he developed equations, using the drop size and the
physical properties of the drop and atmosphere, in order to estimate
the drop axis ratio, the projected horizontal drop diameter and the
terminal velocity.
2.3.2 Rainfall Characteristics
To evaluate soil erosion by rain it is necessary to know about
the rainfall intensity, the duration of the event, the size
distribution of the raindrops at a given intensity, and the kinetic
energy or momentum of the raindrops at a given intensity.
Laws and Parsons (1943) presented the drop size distribution
against rainfall intensity relationship. They used the mean raindrop
size, D50, as the value to represent the particle distribution
Table 4.1 RELEVANT DATA OBTAINED FROM KILIMC AND RICHARDSON STUDY
oo
Kiline's
Run
Number
Rainfall
Intensity, I
(in/hr)
Bed
Slope, S0
(percent)
I
1.25
5.7
V
1.25
10
IX
1.25
15
XIII
1.25
20
XVII
1.25
30
XXI
1.25
40
II
2.25
5.7
VI
2.25
10
X
2.25
15
XIV
2.25
20
XVIII
2.25
30
XXII
2.25
40
Unit
Water Discharge, q Infiltra
Depth, h
(ft)
xlO3
(cfs/ft
width)
xlO4
tion, f
(in./hr)
1.585
2.79
0.4965
1.436
3.16
0.3966
1.212
3.32
0.3534
1.089
3.49
0.3075
0.983
3.59
0.2810
0.906
3.66
0.2618
2.405
7.17
0.3138
1.960
7.27
0.2871
1.748
7.40
0.2554
1.570
7.41
0.2493
1.319
7.45
0.2385
1.198
7.48
0.2304
Kinematic
Viscosity.v
(ft2/sec)
xlO5
Bed Shear
Stress, Tq
(lb/ft2)
xlO2
Sediment
Load, q$
(lb/sec/
ft width)
xlO3
1.282
1.0724
0.096
1.39
2.2212
0.294
1.282
2.5163
0.548
1.22
3.3846
0.644
1.22
4.107
0.922
1.22
4.9101
1.34
1.43
1.845
0.30
1.39
3.8316
1.508
1.43
4.481
2.974
1.39
5.0406
5.686
1.282
5.3696
10.15
1.282
6.0544
13.096
172
D + W sin 0
tan < Slope
W cose L
or
n ^ W(cos 0 sine cot)
1 1 Slope 
(3.18)
(3.19)
Consequently on a horizontal bed (e= 0) erosion must take place
when the drag force exceeds
D  > (3.20)
Hor* (L/D) + cot i
This approach can also be applied to the bed shear stress since
O
D = t A.d which indicates that the correction factor for incipient
ole r
motion due to a longitudinal slope is
Slope
Hor.
' 0 Slope _
I T I Hor.
cose sine cot = cos
tane
ta
(3.21)
This has been proposed by many authors, White (1940), Olivier
(1967), Christensen (1970), Stevens and Simons (1971), Stevens et al.
(1976), Fernandez Luque and van Beek (1976), Smart (1984), Naden
(1987), etc. In addition sediment transport books like Bogardi
(1974), Vanoni (1975), Simons and Senturk (1976), Garde and Ranga
Raju (1977) and Graf (1984) and others, have also included this
approach in their critical bed shear stress or critical bottom flow
velocity relationships for sloping bed conditions. Graf has indica
ted that the experimental test of the incipient motion relationship
369
the new solution. These are very similar to the values obtained
using Equation 3.21 in the original model. Figure B.2 shows the
general agreement of predicted suvalues and the required
suvalues by the proposed method.
Based on the solution obtained from the regression analysis of
Kilinc and Richardson's data and using Equation 3.22 to represent the
longitudinal slope correction factor for the timemean critical bed
shear stress the and suvalues can be obtained from
C =
2.2 x
107
y c tane
1 s
0.78
 2.6 cose
y$ y tani>
h ^ 17$! I Vt y
de 0.54 g de (Y s Y )
ft'
(B.4)
and
su = 0.60 0.156Â£n(v*de/v), (B.5)
respectively.
Finally, it is indicated that this solution shows that even
though the error on the estimate of timemean critical bed shear
stress for longitudinal slopes appear to be significantly affected by
the change of the assumed direction of the buoyant force, the
variation in the prediction of the sediment transport rate may not be
significant. This was already indirectly proved earlier by use of
the DuBoys' bedload transport formula.
LIST OF FIGURES
Figure Page
2.1 Definition Sketch 44
2.2 ErosionDeposition Criteria for Uniform Particles .... 68
2.3 Shields' Diagram for Incipient Motion Including Mantz
Extended Curve for Fine Cohesionless Grains 70
2.4 Typical Velocity Profile and Shear Stress Distributions
for Flow With and Without Rainfall 82
2.5 Surfaces of Detachment, Transport and Maximum
Erosion Rate 126
3.1 Particle Travel Distance 160
3.2 Schematic Saltation Length Approach for Overland Flow . 163
3.3 Effect of Longitudinal Slope on the Saltation Length
of a Grain 167
3.4 Forces Acting on a Grain About to Move for the Horizontal
Bed and Sloping Bed Conditions 171
3.5 Incipient Motion for Uniform Flow Condition Following
Ulrich's (1987) Approach 174
3.6 Correction Factor on the Bed Shear Stress due to
Longitudinal Slope for Ulrich's Coarse Material 176
3.7 Evaluation of the Probability of Erosion 188
4.1 GrainSize Distribution of the Sancfy Soil Used by
Kilinc and Richardson (1973) 201
4.2 $ Versus t': The Data Points 207
4.3 $ Versus T' for Given Rainfall Intensity 208
4.4 $ Versus V^ for Given Rainfall Intensity 209
4.5 Versus 'F for Given Bed Slope 210
xii
10
8
6
2 
1.0
.8
.6
.4
LEGEND
Rainfall
in ./hr
mm/hr
1.25
32
o
2.25
57
3.65
93
4.60
117
.2 .
I 1 1 L.
10 20 30
Bed Slope, SQ (percent)
40
50
Figure 5.4. f 'Values for Given Bed Slope.
0
243
and Shahabian (1977) were reviewed in order to define a possible
expression for su by regression analysis. The information is so
limited and scattered that it does not allow such a regression
analysis.
Therefore, following Chiu's original determination su as a
function of v*de/v alone, the initial expression for su is suggested
su = C5 + C6 n(v*de/v) (4.35)
where
Cg = value of su when v*de/v = 1
Cg = (1/2.3)*(slope of the su versus v*de/vcurve in semiloga
rithmic representation).
Both coefficients, C5 and Cg. may be assumed to be functions of
the remaining parameters related to su.
To test the hypothesis of the validity of Equation 4.35, Chiu's
C^ constant value of 2.2 x 10^ ft~^ was initially used with the in
formation obtained from Kilinc and Richardson in Equation 3.17 in
order to obtain the probability of erosion, p. The n+ was obtained
from the polynomial approximations of the error function using Equa
tion 26,2.23 from Abramowitz and Stegun (1972, p. 933). To use this
approximation the probability of erosion must be between zero and
onehalf (0 < p < 0.5). But, if p is greater than 0.5, the approxi
mation can also be used by changing the value of p to (1 p) and
then the sign of the calculated value of n+ is changed. Knowing the
n+ value the corresponding suvalue was obtained using Equation 3.37.
228
layer are equal to or close to the terminal velocity of a falling
waterdrop. To reach that terminal velocity the raindrop cannot be
intercepted by any object during its free fall. These, as well as
the uniform rainfall intensity over the area, are among the original
assumptions of the proposed model. Knowing these values the rainfall
momentum flux can be evaluated.
The normal component of the momentum flux due to the rainfall
can group all those rainfall variables into a single physical
parameter
Momentum flux = (mass flux)(velocity)
or for a horizontal bed area A0
RMF = 8j(p I A0)Vt (4.20)
where
RMF = rainfall momentum flux
p = density of waterdrops
I = rainfall intensity
A0 = B Ax = surface area in which rainfall is falling
Vt = mean terminal velocity of the raindrops
nd
vt U/nd) ^ Vj
j=l J
Vj = terminal velocity of raindrop of equivalent spherical
raindrop size, Dj
n = number of raindrops per unit area per unit time
140
Tf = 146 S q0,5 (for q < 0.046 m2/min) (2.78a)
Tp = 14,600 S q2 (for q > 0.046 m^/min) (2.78b)
where
Tp has units [kg/(m min)].
Other assumptions made in the development of the ANSWERS model
were
 subsurface return flow and drainage did not produce
sediment erosion
 sediment detached at one point and deposited at another was
reattached to the soil surface
 detachment and reattached sediment required the same amount
of energy as reouired for original detachment, and
 channel erosion (including rills and gullies) was consi
dered negligible
Dill aha et al. (1982) demonstrated the use of ANSWERS to esti
mate sediment yields on construction sites by evaluating several man
agement alternatives available to prevent erosion and their effec
tiveness during the construction development.
Ross et al. (1980) used the original version of ANSWERS (Beas
ley et al., 1980) with his hydrologic model FESHM (Finite Element
Storm Hydrologic Model) in order to model the rainfall and overland
flow erosion. The channel flow erosion was obtained from an approach
used by Chen et al. (1975).
Park et al. (1982b) used the hydrologic component of ANSWERS
and some of the considerations used in the soil erosion component to
develop their soil erosion model. This model included channel
375
Brandt, S., Statistical and Computational Methods in Data Analysis,
Second Edition, NorthHoi land Publishing Company, New York,
1976.
Brown, C. B., "Sediment Transportation," in Engineering Hydraulics,
H. Rouse (Ed.), John Wiley and Sons, Inc. New York, 1950.
Bruce, R. R., L. A. Harper, R. A. Leonard, W. N. Snyder, and A. W.
Thomas, "A Model for Runoff of Pesticides from Small Upland
Watersheds," J. of Environmental Quality, 4(4), 541548, 1975.
Bubenzer, G. D., "Rainfall Characteristics Important for Simulation,"
in Proc. of The Rainfall Simulator Workshop, Tucson, Arizona,
March 79, 1979, U.S. Dept, of Agriculture, Agricultural Reviews
and Manuals, ARMW10, 2234, 1979.
Bubenzer, G. D., and B. A. Jones, "Drop Size and Impact Velocity
Effects on the Detachment of Soils Under Simulated Rainfall,"
Trans, of the American Society of Agricultural Engineers, 14(4),
625628, 1971.
Carter, C. E., J. D. Greer, H. J. Braud, and J. M. Floyd, "Raindrop
Characteristics in South Central United States," Trans, of the
American Society of Agricultural Engineers, 17(6), 10331037,
W/T.
Chapman, G., "Size of Raindrops and Their Striking Force at the Soil
Surface in a Red Pine Plantation," Trans, of the American
Geophysical Union, EOS, 29(5), 664670, 1948.
Chen, C. L., and V. T. Chow, "Hydrodynamics of Mathematically Simu
lated Surface Runoff," Civil Engineering Studies, Hydraulic
Engineering Series No. 18, University of Illinois, Urbana
Champaign, Illinois, 1968.
Chen, C. L., and V. E. Hansen, "Theory and Characteristics of
Overland Flow," Trans, of the American Society of Agricultural
Engineers, 9(1), 2026, 1966.
Chen, Y. H., F. M. Holly, K. Mahmood, and D. B. Simons, "Transport of
Material by Unsteady Flow," in Unsteady Flow in Open Channels,
Water Resources Publications, Fort Collins, Colorado, I,
313365, 1975.
Chiu, T. Y., "Sand Transport by Water or Air," Ph.U. Dissertation,
University of Florida, Gainesville, Florida, 1972.
Chow, V. T., Open Channel Hydraulics, McGrawHill Book Co., New York,
1959.
104
erosion. The most significant ones, in terms of the frequent use,
and the more recent ones are presented here.
2.7.1 Use of Existing Stream Sediment Transport Equations
in Overland Flow
Some studies have considered the use of existing deterministic
or stochastic sediment transport equations to evaluate the soil ero
sion due to overland flow with or without rainfall. These equations
relate the sediment transported by the flow to the bed shear stress,
tQ, bed slope, SQ, flow rate per unit width, q, and particle size, d,
in different ways. However, as indicated in Section 2.5.1.4, the
criteria for erosion in each equation may end up as one of the fol
lowing: a critical velocity or a critical bed shear stress in which
some researchers have used Shields' Diagram (Figure 2.3) to obtain
the timemean critical bed shear stress. Graf (1984) separated the
available sediment (bedload) transport equations based on the criter
ia of erosion used. He named the Schoklitschtype equations to the
equations corresponding to a velocity relationship, and named the
DuBoystype equations to the ones corresponding to a shear stress
relationship. Table 2.3 presents some of the bedload formulas which
represent each group.
The corresponding exponent value of each of the previously men
tioned parameters may change in each equation depending on the as
sumptions and initial conditions from which the equation was devel
oped. In addition, general equations like Manning's equation and
Equation 2.41 for xQ can be used to relate tq to q and SQ. Meyer and
257
of the reported Cs values. The mean value of Cs and the standard
deviation (= error) for each data point are presented on Table 5.2.
Appendix A presents general considerations on how to obtain the
absolute and relative errors of variables related to each other by a
general equation. This approach is used in the following sections in
order to obtain the probable errors of the observed $ and ^ values.
5.2.1 Estimated Relative Error of qs
The volume of sediment discharge per unit width qs is defined
as
qs = q Cs (5.1)
Taking the derivative of this expression yields
9qs = Cs 9q + q 9CS
(5.2)
Dividing both sides by qs definition (Equation 5.1) produces
9qs = 9q + 9cs
qs
(5.3)
Using this equation and the general expression for a relative
error described in Appendix A, Equation A.3, the estimated relative
error of qs may be expressed as
= 5 +
q$ q cs
(5.4)
293
Table 5.11 REQUIRED MINIMUM WATER DEPTH TO USE THE
PROPOSED MODEL
Rainfal1
in/hr
Intensity
(mm/hr)
Water Depth
ft x 10^ (rim)
So =
5.7%
So =
40%
0
(0)
10.9
(0.333)
10.2
(0.310)
1.25
(32)
9.45
(0.288)
8.68
(0.265)
2.25
(57)
8.27
(0.252)
7.50
(0.229)
3.65
(93)
6.61
(0.202)
5.85
(0.178)
4.60
(117)
5.49
(0.167)
4.25
(0.130)
165
3.2.4 Evaluation of Average Saltation Length, l
Chiu (1972) proposed a different formula for the length of sal
tation of a grain than the usually accepted Einstein formula of 1950
(Einstein, 1950). Chiu based his approach on the active forces act
ing on the grain while saltation occurs suggesting that, under the
same hydraulic conditions, a larger grain should be expected to tra
vel a shorter distance per jump than a smaller grain. He considered
that the hydrodynamic lift (L) and hydrodynamic drag (D) exerted on
2
the grain (both proportional to dg ) are increasing the saltation
length while the buoyant weight (W, proportional to dg ) is shorten
ing the saltation length. Hence, Chiu assumed that
where Ci is a constant with dimension of area. The relationship
presented by Einstein was that i = X^g where is a dimensionless
constant, i.e., that the jump length should increase with increasing
grainsize.
It is considered that Chiu's approach is better justified phy
sically and may be used to describe the saltation length of grains in
overland flow with rainfall realistically. The basic change is that
the Ci value will no longer be a constant but must be a function of
the bed slope, water depth and rainfall while the other hydraulic
parameters are kept constant.
As mentioned earlier, flow in open channels usually occurs
under very small slopes and the saltation length can be considered
independent of the longitudinal slope. But for overland flows the
longitudinal slope becomes important. It cannot only control the
overland flow but also the erosion rates. It can be expected that
54
Using this method, Henderson and Wooding (1964) proposed a
series of relationships which allowed calculation of the surface run
off from a sloped bed surface at any location along the bed surface
and at any time. The method can also be used to produce the hydro
graph at any point along the sloped plane.
When the kinematic method is used for watershed modeling, the
watershed is divided in segments with constant slope and the water
flow is routed along the watershed segments (Woolhiser, 1975). Wool
hiser (1969) also used the kinematic approach to model the overland
flow on a converging surface on which the water moved toward a center
point in a radial motion.
Morgali (1970) presented computer solutions to this method and
studied the behavior of the equations for both cases laminar and tur
bulent flows. The variation of the flow regime along the sloped bed
was also considered if rainfall and bed surface conditions were
favorable and enough time for the test was allowed. His hydrograph
results agreed very well with the observations. The only discrepan
cies were observed on the rising segment of the hydrograph after the
inflection point of the rising limb and before the equilibrium flow
was reached at the downstream end of the bed surface. The reason for
this is that the kinematic approach does not predict that inflection
point in the rising limb.
Muzik (1974) tested the kinematic wave method against the in
stantaneous unit hydrograph method under laboratory controlled over
land flow due to rainfall conditions. He concluded that runoff from
309
This gives the expected error of the estimate of around 0.35% of the
mean value of the data. This is small enough to indicate that the
first assumption of the residual analysis is satisfied.
The second assumption of the residual analysis, constant vari
ance, can be examined using the residual plot of Figure 5.9. If the
residuals of the points show a recognizable pattern (i.e., cyclical
pattern, residual increasing as the observed value increases or any
defined not random pattern) the model has no constant variance. Fig
ure 5.9 shows that although there are fewer points on the left side
of the plot, the errors of the points are basically evenly distri
buted over the entire range. This is an indication of constant vari
ance over the entire range, which means that the second assumption of
the residual analysis is also satisfied.
The third assumption is that the errors are normally distribut
ed. The residuals, Equation 5.14, of the points were ordered in mag
nitude and a residual probability was obtained using the expression:
Residual Order Number 0.5 x ^qo (5.17)
Probability ND
Using the standardized residual (Equation 5.13) of each point
the residual probability is plotted on the probability paper as
shown in Figure 5.10. To verify that the errors are normally dis
tributed, the points must be located on or near the straight line
representing the theoretical normal distribution. The data available
229
pÂ¡ = momentum flux factor for the distribution of the rain
drop terminal velocity Vj and size Dj
1 0(1 3
6 = I (ir/6)D.J V.
1 I Vt A0 At j=l j J
Figure 4.7 shows the concepts used here. The rainfall momentum
flux normal to the bed (RMFn) for a sloped bed of width B and with
rain falling vertically (Chen and Chow, 1968) becomes
RMFn = p I Vt B Ax cos2e (4.21)
As it is seen from Equation 4.21, the component of the rainfall
momentum flux normal to the bed decreases by the factor, cos2e as
the slope of the bed increases. Therefore, the rainfall effects on
the saltation length will also decrease as the slope of the bed slope
increases.
Studies of the turbulence in overland flow have shown that
rainfall effects are also diminished by the overland flow conditions
(i.e., Yoon, 1970; Kisisel, 1971). Their studies show that as the
flow Reynolds number, Ref, increases, the turbulence of the over
land flow by itself becomes more predominant up to the point that it
can completely overshadow any rainfall effects. A possible reason
for this observation is that, while the bed slope is contant, the
water depth may increase as the Ref increases. Likewise, the
thickness of the viscous sublayer (6) may decrease as the Ref
increases. The protection created by the water depth increase was
already presented with the use of Palmer's results as the evidence to
support that condition.
300
Table 5.12 DATA POINTS WITH PREDICTED ERRORS LARGER THAN THE
ESTIMATED ERROR OF DATA
Data Point
Slope
percent
Rainfall
(in./hr)
Intensity
(mm/hr)
Reasons for
Possible Error
VI
10
2.25
(57)
Error in $ or v 1
0
XII
15
4.60
(117)
Error in 4'.'
XXIII
40
3.65
(93)
su and pvalues
Yalin (1963)
Yalin (1963)
$ = 0.635 sy X1/2 (1 (l/(ay sy))) *n(l + ay sy)
where
* = qs/((SGs l)g ds3)1/2 x = V((Ys Y)ds}
sy = (x xcr)/xcr ay = 2.45 Xcriy/Yg)0,4
EngelundHansen (1967)
f = 0.05 X25 (0m/v+)2
Yang (1973)
Jtog Ct = 5.435 0.286 Â£og(v$ dgg/v) 0.457 og(v*/v$) +
[1.799 0.409 Â£og(vs dgo/v) 0.314 Aog(v*/vs)] *
Aog((Om Sf/v$) (Ucr Sf/vs))
where
Ucr/vs = 2.05 for v*d50/v > 70
or
0cr/vs = (2.5/Â£og(v^dgQ/v) 0.06) + 0.66 for 1.2 < v^dgg/v < 70
Ct = Total sediment concentration
Source: Simons and Senturk (1977) and Graf (1984)
o
cr>
222
The value of CQ combined with the values of Ap A2 anc* A3 frm
the value of C* for deep flowno rainfall condition considered by
Chiu, in other words
C = Ai a3 = 2>2 x 107 ft2 (4.17)
a2co
Hence the water depth effects on the saltation length are bet
ter expressed by
Co cos e
f(h/de) = 1 _Â£ (4.18)
h/de
This expression indicates that as the water depth increases the
saltation length will gradually increase and approach a maximum con
stant value as the ratio h/de increase. That maximum value is the
one obtained by Chiu corresponding to a constant C* value of
2.2 x 107 ft"2. The expression also indicates that the saltation
length is zero when
h/de = C2 cos e (4.19)
Water depths smaller than this value will be considered as a no
flow condition which also indicates no erosion condition. Rainfall
effects can change the condition of a required initial depth before
soil particles can move. The rainfall effects are discussed next
and a description of how these effects may change the initial depth
condition is presented.
102
Empirical equations were also obtained for each of these mean shear
stresses based on the flow discharge and the bed slope.
The instantaneous shear stress from grain roughness was found
to have the probability density function skewed to the right. For
cases where skewness was not excessive a Pearson Type III distribu
tion fitted the observed distributions but for highly skewed distri
butions, the lognormal distribution was found to be better.
2.6.2 Gully Erosion
As water moves downslope by overland and rills flows, the dis
charge increases. This may lead to further erosion and the develop
ment of gullies. A gully makes a deep cut in the soil surface and
can, in most cases, behave as an ephimeral stream of very short dura
tion. The water depth in gullies is usually deep enough to consider
raindrop impact effects in the flow and erosion characteristics as
negligible.
Similar to the rills, the development of gullies depends on the
flow velocity, the bed slope, and soil characteristics. However,
gullies can also be affected by seepage, water table elevation and
soil type stratification (Piest et al., 1975a, 1975b). These re
searchers found that the wall bank collapse due to cave in and ero
sion at the headcut on the upstream end of the gully were the major
material sources being eroded from the gully itself. These two
conditions are responsible for gully growth. However, it was also
reported that the loose soil and debris accumulated in the gully
10
6 i
8
6
15;
103 ,
1
LEGEND
Rainfall
in ./hr
mnrVhr
1.25
32
o
2.25
57
3.65
93
4.60
117
>
x ii
4 5 6 7 8 910
v*de
Figure 5.18. Relationship Between Cs and v*de/v
(percent)
267
100
80
60
40
20
10
8
6
4
2
1
0 10 20 30 40 50
Bed Slope, SQ (percent)
LEGEND
Rainfall
inyhr
mm/hr
1.25
32
o
2.25
57

3.65
93
*
4.60
117
I 1 1 L.
Figure 5.1. Bed Shear Stress Ratio.
69
The second theory is based on the timemean critical shear
stress, Tcr. DuBuat (Graf, 1984) used this approach during the late
eighteenth century, but it did not become popular until the beginning
of this century when Schoklitsch published his results (Graf, 1984).
Since then, other researchers have used this approach too.
In 1936, Shields (Graf, 1984) used the shear velocity, v^,
which represents a measure of the intensity of turbulent fluctuations
near the bottom boundary. This is related to the bed shear stress by
the expression
(2.43)
Shield used this term in order to describe his wellknown en
trainment motion approach which he presented in Figure 2.3. This
diagram is a graphical representation of the threshold movement of
particles. It was developed from a dimensional analysis for longi
tudinal flows without the influence of raindrops. Shields considered
the disturbing force to be the shear force and assumed that the
resistance of the particle to motion should depend only on the form
of the bed and the buoyant weight of the particle. He studied these
forces for different flow conditions and showed that the threshold
movement of particles could be represented by a single parameter
called the entrainment function, E^ defined as
T
gds(SGs 1)
v dc
= Function (_s) (2.44)
v
u = velocity fluctuation near top of grains on the bed
Vp = waterdrop velocity
Vj = terminal velocity of raindrop with equivalent spherical
raindrop size Dj
Vr = storm runoff volume
Vt = mean terminal velocity of the raindrops
v = v + v' = instantaneous vertical (normal) velocity at a given
location
v = timemean vertical velocity
v1 = vertical velocity fluctuation at a given location
v* = shear velocity
v*cr = c^tieal shear velocity
v*s =
vs = particle fall velocity
vs = fall velocity of particle with size d35
3 3
W = buoyant weight of grain
We = p h 0m2/ r
WS = weight of splashed soil by single waterdrop impact
w = w + w' = instantaneous lateral velocity at a given location
w = timemean lateral velocity
w' = lateral velocity fluctuation
X_ = slope length
x = longitudinal distance
x' = distance measured across the flow from its bank
x* = x/Lo
xcr = critical length to initiate erosion
xx iv
143
The model gave reasonable results when compared with data from
many different conditions considered, even in some cases where only
user's manual information without calibration was used. The CREAMS
model has been used in other areas different from the ones originally
tested, e.g., Foster and Lane (1982) used it to estimate sediment
yield from rangelands.
There are more mathematical models available in the literature
(e.g., Bruce et al., 1975; Khaleel et al, 1979, and others) consider
ing the soil erosion contribution to their water quality, nutrients
removal or pesticides transport models. Most of these models have
adopted existing and already discussed erosion models as part of
their larger scope models.
In addition, there are many more general soil erosion models
like SEDIMOT II (Sedimentology by Distributed Model Treatment, Wilson
et al., 1981; and Warner et al., 1981); KYERMO (Kentucky Erosion
Model, Hirschi et al, 1985) and others available in the literature.
The general approach used in these models was already presented here
(e.g., SEDIMOT II used MUSLE to estimate erosion and KYERMO followed
most of the studies by Foster and Meyer or Foster et al.) and may not
require further discussion. The reader is referred to the original
papers if further details are needed.
47
Robertson et al. (1966) and Yoon (1970) also presented the
momentum equation for the case of steady spatially varied flow over
an impervious surface with mild slope and discussed the significance
of each term of their equation. Both studies used almost the same
assumptions and presented the momentum equation in the form
2
3h (1 Bq ) = S Sf ZBlq + 1 VD cos a (2.20a)
9* 9^ gh2 9h
I T 1 U I 1
Sj = S Sf S2 + S3 (2.20b)
in which B = 1 for Robertson et al. study, and Sj, S2 and S3 repre
sents the simplified form of each term in Equation 2.20a.
Table 2.2 shows the relative magnitude of the terms S, Sj, S2
and S3 with respect to Sf. These values indicate that the most
significant terms of the momentum equation are S = sin6 and Sf and
the remaining terms are at least two orders of magnitude smaller than
S or Sf. The contribution of these less significant terms (S^, S2
and S3) showed fluctuations which were due to the different testing
conditions at the time the measurements were collected (i.e., rain
fall intensity and bed slope).
There are studies of overland flow with rainfall (e.g., Grace
and Eagleson (1965, 1966) and Chen and Chow (1968)) which have indi
cated that the pressure distribution is not hydrostatic. They have
used an overpressure term in the momentum equation in order to
383
Jameson, A. R., K. V. Beard, "Raindrop Axial Ratios," J. of Applied
Meteorology, 21, 257259, 1982.
Jones, D. M. A., "The Shape of Raindrops," J. of Meteorology, 16,
504510, 1959.
Julien, P. J., and M. Frenette, "Modeling of Rainfall Erosion," J. of
Hydraulic Engineering, ASCE, 111(10), 13441359, 1985.
Julien, P. Y., and D. B. Simons, "Sediment Transport Capacity of
Overland Flow," Trans, of the American Society of Agricultural
Engineers, 28(3), 755762, 1985.
Keulegan, G. H., "Laws of Turbulent Flows in Open Channels," J. of
Research, U. S. Nat. Bureau of Standards, 21, 707741, 1938.
Keulegan, G. H., "Spatially Variable Discharge Over a Sloping Plane,"
Trans, of the American Geophysical Union, EOS, 25, 956958,
T547T
Khaleel, R., G. R. Foster, K. R. Reddy, M. R. Overcash, and P. W.
Westerman, "A Nonpoint Source Model for Land Areas Receiving
Animal Wastes: III. A Conceptual Model for Sediment and Manure
Transport," Trans, of the American Society of Agricultural
Engineers, 2ZTFTTnSP1361, 1979.
Kilinc, M., and E. V. Richardson, "Mechanics of Soil Erosion from
Overland Flow Generated by Simulated Rainfall," Hydrology Papers
No. 63, Colorado State University, Fort Collins, Colorado,
1973.
Kinnell, P. I. A., "Splash Erosion: Some Observations on the
SplashCup Technique," Proc. of Soil Science Society of America,
38, 657660, 1974.
Kirkby, A., and M. J. Kirkby, "Surface Wash at the SemiArid Break in
Slope," Z. fur Geomorph. N. F., Suppl. Bd. 21, 151176, 1974.
Kirkby, M. J., and R. P. C. Morgan (Eds.), Soil Erosion, John Wiley
and Sons, Interscience Publication, Chichester, Great Britain,
1980.
Kisisel, I. T., "An Experimental Investigation of the Effect of
Rainfall on the Turbulence Characteristics of Shallow Water
Flow," Ph.D. Dissertation, Purdue University, West Lafayette,
Indiana, 1971.
Horizontal Bed
hi=hH
Sloping Bed
e
i
Theoretical
Bed
h, = hs=hHcose
hj = Initial water depth de= Effective grainsize
hH = Water depth (horizontal bed) 0 = Bed slope angle
hs = Water depth (sloping bed) z h= Depression storage elevation (horizontal bed )
Figure 4.6. Initial Depth Required to Move Grain Under Very Shallow Water Depth Conditions.
III
19.66
1.
VII
18.77
1.
XI
17.11
1.
XV
15.65
1.
XIX
15.26
1.
XXIII
15.29
1.
IV
19.22
1.
VIII
18.77
1.
XII
16.49
1.
XVI
16.49
1.
XX
16.49
1.
XXIV
16.49
1.
3.705
89.09
5.261
92.73
5.897
99.26
6.618
105.32
6.915
108.93
7.304
109.25
4.305
115.57
5.547
119.11
6.631
131.28
6.945
132.29
6.910
132.61
7.361
132.76
37
44
60
77
82
82
41
44
67
67
67
67
Source: Kilinc and Richardson (1973)
1.342
0.203
1.853
0.258
2.469
0.317
2.966
0.360
3.735
0.431
4.264
0.473
1.558
0.312
2.078
0.388
2.782
0.484
3.215
0.544
3.918
0.616
4.611
0.687
.612
0.10300
.432
0.10297
.284
0.10294
.196
0.10288
.069
0.10284
.989
0.10284
.461
0.12982
.309
0.12977
.168
0.12968
.098
0.12968
.002
0.12968
.920
0.12968
1
1
1
1
1
0
1
1
1
1
1
0
358
by Einstein (1950) for open channels or the ones proposed by Foster
and Meyer (1972) for the use of Yalin's bedload formula in overland
flow, might be used as a base for these necessary corrections.
The viscosity of water changes with temperature and with the
sediment concentration. The data used here included the change in
viscosity with respect to temperature but the changes in viscosity
due to sediment concentration were not included. Since the reported
sediment concentration was between 0.5% and 35.5%, some variation on
the viscosity might be expected. This may also affect the settling
velocity of the particle, the Drag coefficient, the saltation length
and the velocity fluctuations on the overland flow with rainfall con
ditions studied here. A study of these variables in shallow water
flows may help in the future to apply the proposed model to the des
cription of the soil erosion process.
The approach to the correction factor of the critical shear
stress on sloping beds presented by Ulrich (1987) seems to have a
good physical basis. However, the disagreement of observed data and
the predicted value by this approach (Equation 3.22) requires further
study. Equation 3.23 may provide the answer to this disagreement.
However, in order to physically show the difference in magnitude,
Equations 3.21, 3.22, and 3.23 need to be tested under very restric
tive conditions not easily found in nature. Specific studies of this
topic are needed due to its importance in the incipient motion cri
terion, design of stable channels, and sediment control on steep
slopes.
14
Musgrave (1947, 1954) presented a review of the knowledge on
sheet erosion and the estimation of land erosion. Using data from
the available literature and from his experiments, he indicated that
the erosion was related to many variables expressed in the following
proportionalities.
Erosion ^ I30^
Erosion ^ S0*33
Erosion XL0,35
Area
where
I30 = maximum amount of rain in 30 minutes of rainfall
(inches)
S0 = slope gradient (percent)
X_ = slope length (feet)
He also presented the relative amount of erosion for different
vegetal covers. Adjustments between studied soils being exposed to
different rainfall, slope and slope length conditions were made in
order to present results of rate of erosion under a common basis. An
example of this procedure was presented in his 1947 study.
Ekern (1953) presented a good summary of the previous knowledge
and information needed about the rainfall properties that affect
raindrop erosion. Then he presented his approach to raindrop erosion
based on the kinetic energy of the natural rainfall and discussed the
45
The literature presents studies in which the continuity equa
tion and momentum equation are used for overland flow descriptions
based on different assumptions and boundary conditions. There are
studies for cases of steady or unsteady state conditions; flows over
porous or impervious surfaces; with socalled physically smooth or
rough boundaries; under laminar or turbulent conditions; with fixed
or loose boundaries, and with or without wind effects. In most stud
ies the momentum correction factor 0 was assumed equal to unity due
to the difficulties in obtaining the velocity distribution of the
shallow overland flow. The use of 0 = 1 assumes uniform velocity
distribution in the cross section. Usually the momentum influx due
to rainfall (last term in Equation 2.19) has been neglected. This
term may be important in cases of steeper slopes or under windy con
ditions (Rogers et al., 1967, and Yoon, 1970). Consequently, the
qualitative judgment of the results of each study must be based on
the assumptions and methodology used by the authors. The possible
general application of the results should also be restricted by the
same considerations.
The study of the relative importance of each term in Equations
2.18 and 2.19 may help to simplify these equations and allow the de
velopment of simple hydraulic models based on these physical princi
ples. Table 2.1 presents the range of values of variables and dimen
sionless parameters in overland flow as reported by Grace and Eagle
son (1965). These values were obtained from an extensive literature
search in order to establish a similarity criterion for the modeling
of overland flow.
113
proposed in later studies. The modified Shields Diagram by Mantz
(1977) is suggested for the case of small size particles (Dillaha and
Beasley, 1983). These latter authors also reported that Davis (1978)
found that the constant 0.635, known as Yalin's constant, had differ
ent values for the case of shallow flow conditions he studied. Davis
found the value 0.88 for sand with a diameter of 342 urn and a value
of 0.47 for coal particles with diameters of 156 urn and 342 pm. How
ever, the original value of 0.635 is still used for overland flow be
cause of lack of additional information (Dillaha and Beasley, 1983).
Dill aha and Beasley (1983) also found that Yalin's equation was
sensitive to the specific gravity and that it was inappropriate as a
transport equation for particles with very small diameters. They
suggested the use of an equivalent sand diameter rather than the ac
tual particle diameter as a way to reduce the effect of change in
specific gravity. The equivalent sand diameters were obtained by
calculating the fall velocity of each particle class based upon their
actual diameters and specific gravities. Then the calculated fall
velocity was used to back calculate a new equivalent sand diameter
using a specific gravity of 2.65. Using this approach the problem of
transport of very small diameters was modeled very easily. In addi
tion, Dillaha and Beasley indicated that this eliminated the problem
of smaller size diameters when the model considered that any particle
with a diameter smaller than 10 pm was treated as washload which the
model routed independently of the larger particles.
168
Qc (1p)
ClA2de A1A3
9(^ 1)
CDde5
1/2
(3.14)
or
where
1/2
(3.15)
C
Ax A
a^T
3
1
This C* includes the effects of bed slope, water depth and rainfall
parameters on the saltation length and has dimensions of inverse area
or inverse length squared.
If Einstein's dimensionless sediment transport rate
(Y$ Y)g de
1/2
(3.16)
is introduced into Equation 3.15 the equilibrium transport condition
yields
JP_ = C. cl/Z d 2 $ (3.17)
1p D e
This equation is the same equilibrium transport equation pre
sented by Chiu with the basic difference that C* was considered as a
true constant while in this study C+ is a function of the water
depth, the rainfall and the bed slope. The value of will be
94
produce sufficient runoff volume to initiate erosion. This could
also be interpreted as a minimum or critical length, xcr, which
depended on the bed slope, runoff intensity (flow discharge per unit
surface area), infiltration capacity and the soil to be eroded. The
rate at which erosion could take place at a distance x from the top
of the sloped bed was also directly related to the runoff intensity,
the downslope distance x, the bed slope angle and a soil erosion
proportionality factor which could account for the amount of material
being removed from a given surface area during certain unit time.
Horton also recognized that either the transporting power of
overland flow or the actual rate of erosion (detachment), whichever
was smaller, could be the governing condition for the overland flow
erosion process. He also related the rill and gully formation on
newly exposed terrains to the critical length required to have over
land flow. He also mentioned that progressive development of rills
and gullies leads to streams and tributary formations that will
define the topography of the drainage basin (including the valleys
and streams). The final shape that he could expect from the original
sloped plane was a concave surface with very small slope gradients at
the base of it representing a delta and then the valley.
The noerosion belt proposed by Horton only considers overland
flow erosion. The raindrop induced transportation (i.e., splash ero
sion) which was previously presented in Section 2.4 of this chapter
may be present in that noerosion belt area but it was not included
in Horton's definition of overland flow erosion.
212
data point number used by Kilinc and Richardson (1973) to identify
their results. The same identification is used in this study.
In Figure 4.2 the data plot almost as a single curve, especial
ly for the data points with $ greater than 0.2. However, observing
the individual values of y1, it can be noticed that for the same y1
value there are more than one corresponding $ data values. Each of
these $ values corresponds to a different rainfall intensity. There
fore, the single curve should not be used. Instead, a family of
curves based on the rainfall intensity can show more clearly the
behavior of the soil erosion process. Figure 4.3 shows these curves
with each of them corresponding to the rainfall intensity used by
Kilinc and Richardson in their study. The curves are almost straight
in double logarithmic representation. This may tempt some research
ers to look for some empirical power relationship between $ and V.
This study does not follow that approach and, as mentioned before,
the main goal is to obtain a physically based description of the ero
sion process.
Figure 4.4 shows how the curves separate more from each other
when the relative flow intensity 'i'1 is corrected by the longitudinal
slope effects and form the new variable using Equation 4.9. With
this new definition the effect of the rainfall on the erosion process
may become more clear. That is to say, for equivalent soil detach
ment conditions (same v'), the rainfall intensity effects will in
o
crease the amount of soil material being eroded. This can be done by
increasing the detachment and transport capacity of the flow condi
tions.
LIST OF TABLES
Table Page
2.1 RANGE OF VALUES OF VARIABLES AND SOME DIMENSIONLESS
PARAMETERS IN OVERLAND FLOW 46
2.2 RELATIVE MAGNITUDES OF THE TERMS S' Si, So and So in
TERMS OF Sf 48
2.3 SOME STREAM BEDLOAD TRANSPORT FORMULAS 105
4.1 RELEVANT DATA OBTAINED FROM KILINC AND RICHARDSON'S STUDY. 193
4.2 DIMENSIONLESS PARAMETERS CALCULATED FROM KILINC AND
RICHARDSON'S DATA 195
4.3 CRITICAL SHEAR STRESS AND DIMENSIONLESS SEDIMENT
TRANSPORT VARIABLES 197
4.4 IDENTIFICATION OF DATA POINTS AND GENERAL LEGEND
FOR FIGURES IN THIS STUDY 211
5.1 ESTIMATED RELATIVE ERROR OF THE DATA 256
5.2 SEDIMENT CONCENTRATION VALUES AND THEIR CALCULATED
RELATIVE ERROR 258
5.3 ESTIMATED ERROR OF THE LONGITUDINAL SLOPE CORRECTION
FACTOR 262
5.4 ESTIMATED RELATIVE ERROR OF $, yJf AND RELATED VARIABLES 264
5.5 DATA POINTS WITH POTENTIALLY LARGE ERRORS OF y' 270
0
5.6 AVERAGE RELATIVE ERROR OF $ AND Cs FOR EACH DATA SET
WITH SAME RAINFALL INTENSITY 279
5.7 AVERAGE RELATIVE ERROR OF $ AND Cs FOR EACH DATA SET
WITH SAME BED SLOPE 279
5.8 DATA POINTS WITH POSSIBLE LARGE ERRORS OF $ 282
5.9 ANALYSIS OF VARIANCE 286
5.10 STATISTICAL ANALYSIS OF THE COEFFICIENTS IN THE MODEL . 289
x
332
the bed roughness and viscosity is very high, which will not allow
the flow to become turbulent enough to detach of soil particles.
Another point of interest about su and the probability of
erosion, p, is the behavior of the error of these parameters in the
prediction of $ when the probability of erosion approaches the value
of unity (n+ in Equation 3.38 approaches to minus infinity). There
are certain physical limitations on the measurement of su that do
allow one to predict the value of su with a better accuracy of
approximately 0.02. As the timemean bed shear stress becomes very
large with respect to the timemean critical shear stress, tq =
100 Tcrs (the probability of erosion will be approaching unity), the
significance of the turbulence random effects represented by su
will become more restrictive in order to predict the real value of p.
This is better explained by studying the effects of the errors
in the prediction of su by Equation 4.41. Table 5.16 presents the
suvalues obtained from Equation 4.41 and the suvalue required
to obtain perfect agreement in the predictions of $ by Equation 4.37.
This table also presents the predicted probability of erosion and the
respective relative error in su and $, which is directly related to
the errors in the rÂ£ values.
The errors in the prediction of su are not very significant
in the errors of $ when the probability of erosion is far from
p = 1.0. The errors of $ are about twice the error of su when the
63
These assumptions lead to the equation
t0 = yti S (2.42)
which some researchers have used as the real value for Tq while
others have used it correctly as a first approximation only.
Dynamic Approach. Another form to obtain t0 is by solving
the dynamic equation (Equation 2.19) for and obtaining from
Equation 2.41. Based on the assumptions made by the authors of each
study, the representation of the dynamic equation may be slightly
different. As mentioned before, Keulegan (1944) was the first to
express that equation for the case of spatially varied flow like the
case of overland flow with rainfall. Other articles which have pre
sented their derivations for this equation or at least have presented
possible methods to solve it are Woo and Brater (1962), Morgali and
Lindsey (1965), Grace and Eagleson (1965, 1966) Ligget and Woolhiser
(1967), AbdelRazaq et al. (1967), Chen and Chow (1968), Chow (1969),
Morgali (1970), Eagleson (1970), Yen and Wenzel (1970), Yen (1972),
Yen et al. (1972), Kilinc and Richardson (1973), and others. The
dynamic equation can only be solved by numerical techniques due to
the complexity of the equation.
Keulegan (1944) recommended that before any approximate solu
tion is attempted the dependence of the friction factor on the flow
Reynolds number is required to be well known. Izzard (1944) was
among the first to present that relationship from curve fitting of
data collected from rougher paved plots. Izzard also obtained that
the water depth was proportional to the cubic root of the
388
Morgan, R. P. C., "Field Studies of Sediment Transport by Overland
Flow," Earth Surface Processes and Landforms, 5, 307316, 1980.
Morgan, R. P. C., Soil Erosion and Its Control, Van Nostrand Reinhold
Co., New York, 1986.
Mosley, M. P., "Rainsplash and the Convexity of Badland Divides," Z.
fur Geomorph. N. F., Suppl. Bd. 18, 1025, 1973.
Mosley, M. P., "Experimental Study of Rill Erosion," Trans, of the
American Society of Agricultural Engineers, 17, 909913, 916,
mr.
Moss, A. J., and P. H. Walker, "Particle Transport by Continental
Water Flows in Relation to Erosion Deposition Soils and Human
Activities," Sedimentary Geology, 20, 81139, 1978.
Moss, A. J., P. H. Walker, and J. Hutka, "Raindrop Stimulated Trans
portation in Shallow Water Flows: An Experimental Study,"
Sedimentary Geology, 22, 165184, 1979.
Mualem, Y., and S. Assouline, "Mathematical Model for Rain Drop Dis
tribution and Rainfall Kinetic Energy," Trans, of the American
Society of Agricultural Engineers, 29(2), 494500, 1986.
Musgrave, G. W., "The Quantitative Evaluation of Factors in Water
ErosionA First Approximation," J. of Soil and Water Conserva
tion, 2, 133138, 1947.
Musgrave, G. W., "Estimating Land ErosionSheet Erosion," Inter
national Association of Scientific Hydrology Pub. No. 36, Tome
1, 207215, 1954.
Mutchler, C. K., "Parameters for Describing Raindrop Splash," J. of
Soil and Water Conservation, 22(3), 9194, 1967.
Mutchler, C. K., "Splash Droplet Production by Waterdrop Impact,"
Water Resources Research, 7(4), 10241030, 1971.
Mutchler, C. K., and C. E. Carter, "Soil Erodibility Variation During
the Year," Trans, of the American Society of Agricultural
Engineers, 2'6', 11021104,"TlBnW.
Mutchler, C. K., and J. D. Greer, "Effect of Slope Length on Erosion
from Low Slopes," Trans, of the American Society of Agricultural
Engineers, 23(4), 866869, 876, 1980.
Mutchler, C. K., and L. M. Hansen, "Splash of a Waterdrop at Terminal
Velocity," Science, 169, 13111312, 1970.
129
Dc = detachment capacity of flow [weight/(time unit area)]
Tc = transport capacity of flow [weight/(time unit width)]
The net deposition was indicated by negative values of Dp
while positive values of Dp indicated net soil removed from the
location or detachment. This rate of flow detachment was assumed to
be a function of the difference between the actual sediment load and
the flow transport capacity or
(2.71)
F = af (Tc 9S)
which from Equation 2.70 gives
(2.72)
The "af" coefficient was indicated to have different values
for detachment and deposition and later was presented as the ratio
between a coefficient for soil detachment (to consider soil resis
tance to erosion by flow) and a coefficient for soil transportation
by flow. This ratio was obtained based on the consideration that
Dc and Tc were functions of the tractive forces of the flow
expressed by the boundary shear stress to the threehalf power,
They considered that the rainfall detachment rate, DÂ¡, in
cluded contributions from both rainfall detachment and transport.
This term also reflected the combined capacity of raindrop impact and
the sheet flow between the rills to detach and transport particles to
the rills. Other general considerations or assumptions made by the
authors were: uniform quasisteady flow, and use of the Chezy equa
tion to determine the water depth. With this information and the
68
Figure 2.2. ErosionDeposition Criteria for Uniform Particles
(after Hjulstrom, 1935; Graf, 1984; reprint with permission of Water
Resources Publications).
500
184
grain size ratio only. For the case of overland flow with rainfall,
the raindrop impacts can increase the X and  T0  values for an
instant at the point of impact and in the surroundings. The values
of X and  tQ  may change as the turbulence generated by the raindrop
impact dissipates as a function of time and distance from the impact.
2
Since both terms are function of u^ at the location in consideration
the (X/  t0  ) ratio is expected to be constant. It is considered
that the area affected by the raindrop impact per unit time is very
small compared to the total surface area and the same unit time in
consideration. In addition, individual raindrop impact effects last
only an instant. Consequently, the assumption of the (X/  T0  )
ratio being only a function of the roughness to grain size ratio can
be accepted for the case of overland flow with rainfall. The error
which this assumption may introduce is also considered to be within
the accuracy with which the (X/  T0  ) can be measured in open chan
nels.
It should be mentioned that Harlow and Shannon (1967), Wenzel
and Wang (1970) and Huang et al. (1982) have shown by numerical anal
ysis that the bed shear stress increases instantaneously at the point
of impact and on the surroundings. Unfortunately, none of these
studies considered overland flow over the solid surface and the solu
tions obtained were for smooth surface with or without a water layer
over it.
154
The rainfall intensity is assumed constant over the area. The
rainfall has known raindrop distribution, momentum flux, and kinetic
energy. No consideration on how these rainfall characteristics
change with time and space is made in this study. The raindrop paths
are assumed to be completely vertical. No wind effects on rainfall
or flow are considered.
Hydrological considerations like the interaction between preci
pitation, evaporation, seepage and overland flow are not considered
in this study. Seepage and wave forces are not considered in this
study either.
It is assumed that the soil is saturated and overland flow is
present over the area in consideration with at least a minimum water
layer over the soil particles.
3.1.2 Major Considerations and Assumptions
As it was presented by Chiu (1972), this model assumes that the
basic mode in which particles move is considered to be the motion by
saltation. Thus, rolling and sliding are very low saltations and
suspension is a very high (or infinitely long) saltation. The salta
tion concept may also be applied in the case of raindrop splash ejec
ting soil particles out of the water layer. The splashed particles
will travel a longer distance in air than they would do in water if
they had remained in the water layer. This is because the splashed
grains moving in air usually may be subjected to lower flow resis
tance than when they are moving in water. However, the splashed
particles will return to the water layer at other locations. The
200
should be. Figure 4.1 shows the grain size distribution of the ori
ginal soil and the approximations made to limit the real unknown max
imum and minimum grain sizes of the grain size distribution. These
approximations simplify the computation process and the error intro
duced into de is not going to be as big as the errors introduced by
uncertainties of the sieve analysis.
4.2.3 Drag Coefficient Evaluation
Cq values can be obtained from established curves, published
in practically every fluid mechanics book, but the ones most fre
quently presented are the ones from Rouse (1946). To obtain the Cq
value for a given grain size it is required to know the settling ve
locity of a grain of that size (vs). This velocity can be obtained
using Equation 3.4 already presented in the previous chapter. Assum
ing the soil grains are spherical with size equal to de, Equation 3.4
still has two unknowns, v$ and Cq. The solution can be obtained by
trial and error using Cq values from the Cq curves and solving for
vs. With this vs, the Cq value is verified evaluating first the
settling Reynolds number
Res = de Vs (4.2)
and then obtaining the corresponding Cq value. The process is
repeated until the numerical variation of Cq is very small.
To make this process more systematic, it was decided to use one
of the already existing mathematical approximations. The equation
presented by Olson (1973), also available in Graf (1984), was select
ed for this study because of its simplicity, it is not difficult to
be used in an iterative process, and gives good results in the range
of Res values the data have. Olson's equation
101
A study on the hydraulics of flow in a rill was recently pre
sented by Foster et al., 1984a, 1984b. For their experimental condi
tions (Foster et al., 1984a), the mean velocity was only slightly af
fected by rainfall. It was reported that, even though the flow was
fully turbulent, the friction factor decreased with increasing flow
Reynolds numbers showing a laminarlike behavior in special for the
lower flow rates. The energy and momentum coefficients varied with
section location along the rill and discharge, and typical values
were reported to be around 1.60 and 1.20, respectively. The longi
tudinal velocity profiles were logarithmic as indicated by the
Prandtlvon Krmn equation but with the empirical coefficient < of
the equation varying from 0.2 to 1.3 depending on the location along
the rill and the flow discharge. The general tendency of < was to
increase as the discharge rate increased.
In Foster et al. (1984b) the shear stress values in a rill were
presented but this time the measurements were obtained under no rain
fall conditions due to the interference of raindrop impacts with the
shear stress measurements. The shear stress around a cross section
and along the rill was highly irregular mostly because of the highly
turbulent nature of the flow. Their instrument measured the time
mean shear stress (Tg) due to the grain roughness (roughness due to
grain size; Graf, 1984). The total time mean shear stress (x0) was
obtained using = yR'Se (Equation 2.41) and the timemean shear
stress (Tf) due to form roughness (roughness due to bedforms; Graf,
1984) was obtained by the equation
337
The probability of erosion, p, has been assumed to be found
from the normal distribution curve. This distribution is symmetric,
which could lead one to believe that the same growing error of $ can
occur as p approaches zero. Fortunately, the 1 p values
approach a finite value (zero) as the value of p approaches zero.
Therefore, errors of $ due to errors in p ^ will approach a
finite minimum value as p approaches zero. $ and jB will also
approach zero.
So, the real problem is when the denominator 1 p approaches
zero as p approaches unity. This makes ^ P ^ tend toward plus
infinity and very small changes in p will significantly increase the
error of
1 p
To this writer's knowledge, neither Einstein (1950) nor Chiu
(1972) indicated any problem with the relationship ^ P in the eval
uation of predicting values of $. However, the problem of their
equations as the probability of erosion approaches unity will be the
same as the ones discussed here. The general explanation given for
Einstein's formula being in error at high ^values is that at low
Tcr/T0values the experimental data included suspendedload material
and Einstein's equation is for bedload transport only (Graf, 1984).
Chiu considered his equation as representative of the total load
transport given that the bed material size represented by the effec
tive grainsize does not contain very fine material.
The only correction which has been proposed to the Einstein
bedload function at high sediment rates is the use of a function to
0.5
2
3 4 56789 10
0.4
0.3
0.2
0.1
0
2 3 4 56789 10
v. de/v
v d /
* e/v
ro
cn
CJ
Figure 4.11. Predicted and Required suValues
1 p
Figure 5.15. Changes in p/(l p) Due to Errors in p Evaluation.
CO
CO
O'
tan 0 %
Figure 3.6. Correction Factor on the Bed Shear Stress due to Longitudinal Slope for
Ulrich's Coarse Material.
36
resulting breakdown of the structure of the aggregates in the soil by
the raindrop impacts.
DePloey and Savat (1968) used autoradiographies of radioactive
sand to study the splash mechanism. Their results showed the impor
tance in the splash phenomenon of the grainsize distribution of
sands, the slope gradient, the angle of ejection, the distribution of
grains around the point of impact of the raindrops, the characteris
tics of the rain, and the physical properties of the soil. Using
their data and physical considerations in developing a mass balance
of the soil particles, they were able to describe the splash mechan
ism for horizontal surfaces, sloped surfaces, and for segments of a
convex slope.
Morgan (1978) indicated that his results of rainsplash erosion
from field studies of sandy soils confirmed the relationships between
splash erosion, rainfall energy, and bed slope obtained in laboratory
experiments by other researchers. He also reported that only 0.06%
of the rainfall energy contributed to splash erosion and that the
major role of the splash process is the detachment of soil particles
prior to their removal by overland flow.
Poesen (1981) studied the erodibility of loose sediments as a
timedependent phenomenon. He indicated that the variations in the
detachability of soil particles during the rainfall event could be
explained by changes in water content (including the liquifaction and
the development of a water layer on the surface), cohesion and granu
lometric composition of the top layer. In his case the presence of
217
f(e) = function to represent the longitudinal bed slope effects
on the saltation length
m = exponent to be evaluated
The value of m is expected to be near unity, however, it may be
different since the tangential component of the buoyant weight (Wsine)
has not been implicitly included as an additional force able to in
crease the saltation length. This component is explicitly included
in the development of the critical shear stress correction factor due
to bed slope (Equation 3.25).
The slope effects mentioned here are based on deep flow with no
rainfall conditions. It is recognized that as the bed slope in
creases the water depth will decrease. That may cause additional
effects on the saltation length. These effects are discussed separ
ately in the following paragraphs.
Let the depth of the water flow be gradually decreased and as
sume the remaining parameters are considered constant (i.e., lift,
drag, and buoyant weight). When the water depth is of the order of
magnitude of the potential maximum elevation a saltating grain can
reach, the saltation length is going to be affected. It is recog
nized that the maximum elevation a sand grain can commonly reach in a
liquid fluid is a few times its grain size. This means that surface
tension effects become important at low water depths while they may
be neglected completely in deep water. The surface tension will try
to keep the grain inside the water layer thus reducing the available
313
Due to lack of more information and data, nothing else can be
concluded about the independence of the errors, except that, as a
group, the 24 points show a generally random residual. The residuals
are well distributed on the range of ^values and are probably inde
pendent (see Figure 5.9).
5.4 The Saltation Length Process and the Values
The saltation of grains in fluids is a complex process. The
proposed model followed the approach suggested by Chiu (1972) for
open channels and modified that approach to be applicable to low
depth overland flow on slopes with rainfall. The three major varia
bles necessary to add to the original approach are water depth, bed
slope, and the rainfall intensity. The effects of these three varia
bles on the saltation length were initially treated as independent
effects acting simultaneously. The three basically independent equa
tions (Equations 4.13, 4.18, and 4.27) were multiplied by each other
to represent the saltation length.
Further physical considerations and results from the evaluation
of the estimated coefficient values led to the elimination of Equa
tion 4.27 and the modification of Equation 4.18 into Equation 4.28.
This new equation has combined the effects of all three parameters
into a single expression indicating that the effects of the bed
slope, water depth, and rainfall intensity on the saltation length
are closer correlated to each other than was originally thought.
Therefore, the water depth correlating function required the addition
321
Therefore, the average distance traveled by the grain in each jump
will approach zero as the bed slope angle approaches the angle of
repose of those grains. In this case the grains can roll practically
an infinite distance or at least up to a location where the bed slope
diminishes and the hydrodynamic forces cannot move the grain again.
It should be noted that as the inclination of the bed approach
es the angle of repose, the Tcrs value must approach zero. There
fore, the probability of erosion will approach unity, which means all
grains on the surface area in consideration will be subject to mo
tion. In other words, the topmost layer of grains may move without
any hydrodynamic force. If there are no hydrodynamic forces applied
to the grains, the saltation length as expressed by Chiu would have
to approach zero (rolling or sliding induced by gravity only). This
is the purpose of Equation 4.13.
Here, in order to simplify the mathematical representation of
the rolling or sliding of grains, the saltation length is said to be
equal to zero because the grain is in contact with the bed surface
all the time. This can occur on physically "smooth" surfaces which
have negligible roughness. For normal cases, the rolling of grains
over a bed surface composed of similar grains may show very small
separations (jumps) of the rolling grain from full contact with the
bed grains. This very short saltation can be represented in Equation
4.13 by a coefficient C7 and f(e ) being expressed as
Eroded Grains pÂ¡na
Originating Area Deposition
FLOW
Figure 3.1. Particle Travel Distance.
72
= ad To(To Tcr^
(2.45)
where
qs = volume of sediment transported per unit width per unit time
ad = constant
Shields' diagram has been used by other researchers who have
developed soil erosion models for small watersheds and field plots.
The studies presenting these models are discussed later in this chap
ter. It should be recalled that the Shields' diagram is based on
experiments with cohesionless materials, a fact which researchers may
have recognized, although some of the soils they studied did have
some cohesion properties.
Smerdon (1964) also used the DuBoys' equation to calculate the
critical tractive force required to erode rough channels with shallow
flows under rainfall conditions by measuring the sediment load and
obtaining tq from measurements of bed slope, water surface slope,
water depth and flow rate. Results indicated that TCr changed
very little as a result of intense rainfall on the shallow flow. In
general, tcr values increased slightly when rainfall was applied
and the suspended particles concentration were reduced due to the re
duction of the flow velocity. The great velocity reduction was ob
served at test runs with the shallower water depths. He also recog
nized that the reported shear stress in his study were timemean
values and possibly much less than the instantaneous values which
actually produce erosion. His results were considered good for the
377
Davis, S. S., "Deposition of Nonuni form Sediment by Overland Flow on
Concave Slopes," M. S. Thesis, Purdue University, West
Lafayette, Indiana, 1978.
Deming, W. E., Statistical Adjustment of Data, John Wiley and Sons,
Inc., London, England, 1943
DePloey, J., and J. Savat, "Contribution a 1'etude de 1'erosion par
le splash," Z. fur Geomorph. N. F., 12(2), 1968.
DePloey, J., J. Savat, and J. Moeyersons, "The Differential Impact of
Some Soil Loss Factors on Flow, Runoff Creep and Rainwash,"
Earth Surface Processes, 1, 151161, 1976.
Dill aha, T. A., Ill, and D. B. Beasley, "Distributed Parameter Model
ing of Sediment Movement and Particle Size Distributions,"
Trans, of the American Society of American Engineers, 26(6),
17661772, 1777',' 1983.
Dill aha, T. A., Ill, D. B. Beasley, and L. F. Huggins, "Using the
ANSWERS Model to Estimate Sediment Yields on Construction
Sites," J. of Soil and Water Conservation 37(2), 117120, 1982.
DuBoys, M. P., Le Rhone et les Rivieres a Lit Affouillable, Mem. Doc.
Annales de Pont et Chaussees, Series 5, Vol. 18, 1879.
Duley, F. L., "Surface Factors Affecting the Rate of Intake of Water
by Soils," Proc. of Soil Science Society of America, 4, 6064,
1939.
Eagelson, P. S., Dynamic Hydrology, McGrawHill, Inc., New York,
1970.
Einstein, H. A., "The Bed Load Function for Sediment Transportation
in Open Channel Flow," U. S. Department of Agriculture Technical
Bulletin No. 1026, 1950.
Einstein, H. A. and F. M. AbdelAal, "Einstein BedLoad Function at
High Sediment Rates," J. of the Hydraulics Division, ASCE,
98(Hy1), 137151, 1972
Einstein, H. A. and ES. A. ElSamni, "Hydrodynamic Forces on a Rough
Wall," Reviews of Modern Physics, 21(3), 520524, 1949.
Eisenlohr, W. S., Discussion of "Spatially Variable Discharge Over a
Sloping Plane" by G. H. Keulegan, Trans, of the American
Geophysical Union, EOS, 25, 959, lTJi^
Ekern, P. C., "Raindrop Impact as the Force Initiating Soil Erosion,"
Proc. of Soil Science Society of America, 15, 710, 1950.
26
surfaces with different water layer thicknesses over the bed surface.
The variation in splash characteristics with respect to changes in
the water layer thickness was visually explained in those photo
graphs.
But it was not until the late 1960s that the interest on the
waterdrop splash process and splash sequences were really studied
thoroughly. Mutchler authored and coauthored a series of articles
in which the individual characteristics of the waterdrop splash were
presented.
Mutchler (1967) studied the waterdrop splash at terminal velo
city over different types of surfaces with and without a water layer
covering it. He studied the effects of the drop diameter, the water
depth, the roughness and the softness or hardness of the solid sur
faces on the splash characteristics. A set of parameters were estab
lished to describe the geometry of splash. For this he used the
width of the crater of the splash, the height of the splash sheet
wall, the radius of curvature of the splash sheet wall, the angle at
which the sheet wall goes with respect to the water surface, and the
angle at which the splash droplets are ejected from the splash sheet
wall. Since these parameters changed their values with respect to
time he used the characteristic shape occurring at the time of maxi
mum sheet wall height to show the effect of the water layer depth on
the splash. He concluded that the water depth had its greatest ef
fect on the waterdrop splash at depths of about onethird of a drop
diameter and that the splash geometry changed very little at water
depth greater than one drop diameter.
Table B.2 STATISTICAL ANALYSIS OF THE COEFFICIENTS IN THE MODEL
Used for
Evaluation
Coefficient of
Estimated
Value *
EV
Standard
Error of
Estimate
SEE
x 100
(percent)
Student's
tValue**
aj = 0.05
Check Null
Hypothesis
H0 = 0**
C5
s
u
0.60
0.03364
5.6
17.834
Rejected
C6 j
(Equation 4.41)
0.156
0.01890
12.1
8.254
Rejected
m
r
0.71
0.05732
8.1
12.387
Rejected
C2
L*
(Equation B.4)
2.6
0.32396
12.5
8.026
Rejected
C4 J
17
9.13851
53.8
1.860
Not rejected
*Rounded values. Some error can be reduced if more significant digits are used. However, the
unknown accuracy of data does not support the use of this approach.
**For 19 degrees of freedom and 5% level of significance, the Student's tvalue is
t(l9 0.975) = 2*093 (Brandt, 1976).
237
the ratio between the vertical component of the rainfall momentum
flux per unit area (p I Bj Vt cos2fl) and the buoyant weight of the
soil particles per unit area (f(en) (ys y )ds) or
f (I)
p I 8j Vt cos2fl
f(e) *)de
(4.31)
where
fi = angle of inclination of falling raindrops with respect
to the vertical axis
f(en) = function based on the degree of compaction and particle
pattern of the soil (Christensen, personal communica
tion, 1984).
f(en) = 0.685(1 + enr415
en = void ratio of the soil
One of the assumptions of this study is that raindrops are
falling vertically, hence cos2 0*1. As mentioned before, the
porosity of the sandy soil used by Kilinc and Richardson was 43%.
This corresponds to a void ratio of 0.754. Substituting this value
in the f(en) definition gives
f(en) = 0.54
It was mentioned before that for constant rainfall intensity
and raindrop distribution the value of Bj is constant and assumed to
be unity in this study. Now Equation 4.28 becomes
145
or aggregates present on the soil. This can create a seal at the
soil surface because the raindrop impact pressure exerted toward the
soil can compress the top soil layer creating a crust (about 1 to 3
mm thick) which can reduce the infiltration capacity of the soil (Du
ley, 1939; McIntyre, 1958a). However, the presence of this surface
crust is time dependent. During the rainfall event, it can break
down and form again more than once (McIntyre, 1958b; Farres, 1978).
The splash erosion is reduced when the crust is present, but the
increase in runoff or further drop impacts can break the crust and
increase the soil erosion. Ellison and Slater (1945) recognized that
variation of the permeability with time and suggested that their
splash erosion equation would require an additional term to account
for it. In their study, the four principal factors affecting infil
tration rates of the soil studied were: duration of the rainfall,
aggregation of the soils, clay content of the soil and amount of soil
carried away by the raindrop splash.
Lemos and Lutz (1957) reported that the surface crust had a
much greater bulk density, a higher percentage of particles smaller
than 100 pm in diameter, and a lower degree of aggregation than the
underlying soil. The permeabilities of the soil crust may be many
times lower than the permeability of the underlying soil. Tackett
and Pearson (1965) reported values five times smaller while McIntyre
(1958a) reported permeability changes from 106 cm/sec to 103 cm/sec,
respectively. McIntyre's (1958a) results also agreed with Duley's
(1939) observations.
147
Lyles et al. (1969) found that there were significant interac
tions among the variables they studied (clod size and density, rain
fall intensity and duration, and wind velocity). Their study indica
ted that high rainfall intensities of short duration (i.e., 10 min
rain at 56.1 mm/hr (2.2 in./hr)) were as destructive as lower rain
fall intensities of longer duration (90 min rain at 16 mm/hr (0.63
in./hr)) even though the total volume was about 2.5 times larger in
the latter case. It was also reported that wind driven rain was more
effective in disintegrating the clods. Up to 66% more soil was re
moved when winds of 13.4 m/sec (44 ft/sec) were present than for the
no wind condition with all the remaining variables being the same in
both cases.
Like in other earlier studies Lyles et al. (1974) reported that
soil detachment, in this case detachment from clods, decreased with
increased amounts of mulch cover. Their study indicated that air
dried clods showed more soil detachment than the fieldmoist clods.
This effect was associated with the rate at which additional water is
absorbed. They indicated that fieldmoist clods absorb water slower
due to the high degree of saturation while airdried clods absorb
water rapidly and in excess of the liquid limit. This makes the soil
expand and lose some of its shear strength, making it more suscepti
ble to detachment. This stuc(y also showed that the determination of
mulch cover requirements for different levels of soil protection
required the additional consideration of the probability of wind
accompanying rain. On the average, their study showed that a wind of
CHAPTER V
DISCUSSIONS AND MODEL VERIFICATION
5.1 Introduction
This chapter considers many points of the proposed model which
may need further attention. Since it was not possible to find more
data to corroborate the model, it is supported by statistical analy
ses of the observed data and of values predicted from the model. The
effect of the estimated values of the coefficients on overland flow
with rainfall is also presented.
5.2 Error Analysis of the Data
Kilinc and Richardson (1973), give very little information
about measurement errors and accuracy of their data. With the excep
tion of the sediment concentration, Cs, all of parameters reported
for the individual data points are single valued. This does not
allow determination of the possible error of each value. However, a
rough estimate of the possible error of each parameter may be
obtained from typical errors of similar measurements with the same
instrumentation. The results of such an accuracy stuc(y are presented
in terms of $ and Using the definition of $ given by Equation
3.16 and the definition of v' by Equation 4.9, the parameters and
u
their presumed relative error are presented in Table 5.1.
The error of the sediment concentration was calculated from
Csvalues collected at different times during the onehour runs.
The error on C$ is assumed to be equal to the standard deviation
255
272
Bed Slope, SQ (percent)
Figure 5.2. T0Values as Calculated By Kilinc
and Richardson (1973).
357
but studies of the raindrop impact in flowing water (i.e., overland
flow) over a mobile bed are needed.
The transport of particles in this very shallow overland flow
appears to be by very small saltations or rolling. The raindrop
impacts provide additional energy to detach soil particles. Some of
these particles are ejected out of the water flow and travel through
the air before reaching the water layer again. The distance traveled
by each particle and the proportions of particles which travel by air
to the particles which move in the water after the raindrop impact
are needed in order to better estimate the average saltation length
in overland with rainfall conditions. An expression for that average
saltation length might consist of two terms (i.e., saltation length
in water and saltation length in air). However, other effects such
as the ones considered in this study (i.e., water depth, rainfall in
tensity and bed slope effects) must be included and studied more com
prehensively with future data.
The effective grainsize, de, simplifies the use of the sedi
ment transport equation because it only requires one de calculation
to represent the whole grain size distribution. It also requires a
single calculation of Cq over the bed surface. However, the armor
ing of the bed surface or the transport of nutrients and pollutants
in the overland flow may sometimes require the prediction of grain
size distribution of the transported material. A modification of the
proposed model may give that information if additional data are ob
tained and the necessary corrections are made. The concepts followed
APPENDIX C
CONVERSION FACTORS
Quantity
SI Unit
English Unit
Conversion Factor
Mass
kilogram
(kg)
slug
1 slug = 14.5939 kg
Length
meter (m)
foot (ft)
1 ft = 0.3048 m
Time
second (s)
second (sec)
1 sec = 1 s
Area
m2
ft2
1 m2 = 10.764 ft2
Volume
m3
ft3
1 m3 = 35.315 ft3
Velocity
m/s
ft/sec
1 ft/sec = 0.3048 m/sec
Acceleration
m/s2
ft/sec2
1 ft/sec2 = 0.3048 m/s2
Force, Weight,
Momentum Flux
N =
(kg*m)/s2
lbf = lb
1 lb = 4.4482 N
Pressure
Pa = N/m2
lb/ft2
1 lb/ft2 = 47.88 Pa
Energy, Work
J = Nm
ft'lb
1 ft'lb = 1.3558 J
Power
W = J/s
(lbft)/sec
l(lb*ft)/sec = 1.3558 W
Density
kg/m3
slugs/ft3
1 slug/ft3 = 515.4 kg/nv
Dynamic
Viscosity
kg/(m* s)
slug/(ffsec)
1 slug/(ffsec) =
47.88 kg/(m's)
Kinematic
Viscosity
m^/s
ft2/sec
1 m2/s = 10.76 ft2/sec
Unit Weight
N/m3
lb/ft3
1 lb/ft3 = 157.09 N/m3
371
258
Table 5.2 SEDIMENT CONCENTRATION VALUES AND THEIR CALCULATED
RELATIVE ERROR
Run
Sediment Concentration, C$
TimeMean (ppm)
Relative Error*
Percent
I
5436
8.15
V
14483
15.67
IX
24434
5.26
XIII
28493
5.08
XVII
37975
7.29
XXI
44149
8.64
II
6593
5.45
VI
32004
11.52
X
57418
7.77
XIV
103891
9.10
XVIII
172744
19.40
XXII
207585
16.85
III
8054
2.42
VII
43300
8.59
XI
79895
10.51
XV
154167
9.23
XIX
217769
6.87
XXIII
313749
3.25
IV
14344
5.00
VIII
52279
2.34
XII
109234
2.35
XVI
202717
7.55
XX
255279
7.69
XXIV
355885
4.93
Relative error obtained by dividing the standard deviation of C$ of
that run by the corresponding timemean Csvalue.
y/h y/h
Figure 2.4. Typical Velocity Profile and Shear Stress Distributions for Flow With and
Without Rainfall (based on Yoon, 1970).
87
5 a (i) (i)2 (254)
umax max h h
With this equation and with its derivative they developed expressions
for the discharge per unit width q = (2/3)umaxh = I^x, the longitu
dinal mean velocity 0 = (2/3)u , or if the DarcyWeisbach equation
m max
(Equation 2.27) is used
0m = Cg/(3V) Sf (I^x)2]1/3 (2.55a)
Assuming that raindrop effects could make umax at y = (2/3)h
with u = 0 at y = 0 and = 0 at y = (2/3)h they presented the mean
dy
velocity profile for the rainfall induced retarded overland flow which
gave a mean velocity of
m = Cg/(4v) Sf (I*x)2]1/3 (2.55b)
This equation indicates that the mean velocity could be reduced by
about 10%. This was only done in order to indicate the magnitude of
the change between the velocity profile without raindrop effect and
the velocity profile with raindrop effect (but under very specific
conditions).
2.5.1.6 Turbulence
Many studies mention that raindrops striking overland flow in
creases the turbulence in the flow but there are only a few studies in
which the turbulence intensity in overland flow with rainfall have
been quantitatively studied. The local instantaneous velocity of a
particle fluid in a given location is usually expressed in three
component directions, e.g. longitudinal direction, normal direction,
271
possible to determine if all four points or only some of them have a
significant error of
Other data points with possibly significant error of ^ may be
identified in Table 4.1 or Figure 5.2. The value ofT 0 for run XX
(S0 = 30%) is less than the value of T0 for run XIV (S0 = 20%)
even though the rainfall intensity is the same (I = 4.60 in./hr, 118
mm/hr). The expected opposite condition is demonstrated by the re
maining data points. The observed lower value of T0 in run XX can
not be explained from the data of the original study. Other points
in Figure 5.2 show some possible error on T0, but they were pre
viously indicated in the discussion of Figure 4.5 and Vq.
Figures 5.3 and 5.4 show the change of and J'q respectively
for given bed slope. These figures show that the constant rainfall
intensity curves follow the same general pattern. They also show the
previously mentioned data points with possible large errors of Vq.
Both sets of curves show a substantial dependency on the bed slope.
This is to be expected since the bed slope has a direct effect on the
bed shear stress, and thereby on v1 and Yq (Equation 4.8 and Equation
4.9).
The difference between these two figures is the bed slope cor
rection factor on the critical shear stress (Equation 3.25 or Equa
tion 4.9). However, this correction also helps to explain why the
erosion increases as the bed slope increases. In Figure 5.2, is
34
experimental data or statistical foundations to support the use of
their rainfall parameter in their equation. Rose (1960) justified
the use of rainfall momentum per unit area and time instead of using
the kinetic energy per unit area and time. Meanwhile, Gilley and
Finkner (1985) presented statistical analysis which indicates that
the kinetic energy times the drop circumference is better. Apparent
ly the literature shows that there is a majority of studies prefer
ring the rainfall's kinetic energy more than the rainfall's momentum
for the development of their splash erosion equations, but the use of
any of these two rainfall parameter must be physically justified in
each case.
Bubenzer and Jones (1971) also studied the effects of drop size
and impact velocity on the splash detachment. They found that small
er drops produced less splash than the larger ones even though the
kinetic energy, the total rainfall mass and impact velocity were
almost constant. Therefore, more parameters are needed to describe
the splash erosion.
The effect of the bed slope is also very important in the
splash erosion (e.g., Ekern and Muckenhirn, 1947; Ekern, 1950; Free,
1952; DePloey and Savat, 1968; Savat, 1981, and others) because the
soil downslope transport increases as the bed slope increases. Free
(1952) also indicated that the effect of the slope in relation to the
direction of the storm was important in determining the amount of
soil removed from the soil pans. Losses from pans facing the direc
tion of the storm were found to be three times those from pans facing
127
relationships of rainfall and runoff with the soil erosion, were
proposed for each subprocess. For the case of irregular slopes they
considered increments of areas, AÂ¡, with the same slope which
allowed them to present the general equations as
Soil
detachment
by rainfall
DI = SDI AI Â¡2
(2.65)
Soil
transport
by rainfall
TI = sTI S I
(2.66)
Soil detachment by runoff
F = SDF AI 7[(So 1 Q)START + (So ^ Q)END1 (2,67)
Soil transport by runoff Tp = Syp Sg Q5/^ (2.68)
where SDI, Sjj, SDp, and Syp are constants which include the soil
properties effect, and Q is the rate of flow at a given location.
The sediment load carried away from each increment section was consi
dered to be the smaller value of the sediment load from the previous
increment plus the detachment of that increment, or the transport
capacity from that increment. Net erosion (or deposition) for any
increment was obtained from the difference between the sediment loads
entering and leaving the increment.
Meyer and Wischmeier recognized that their relationship for
each process required improvements and some additional parameters
would have to be added to describe the erosion process. It should
also be mentioned that this study was the first one in which Elli
son's subprocesses were applied to a soil erosiondeposition model
202
1/2
Cn = (1 *" ) for Re,. < 100 (4.3)
Res 16 Res
can be introduced into Equation 3.4 and substituting vs from Equa
tion 4.2 definition the expression for Res is presented as
Res
Rde3 1
18v2 (1 + 0.1875 Res)1/2
(4.4)
This equation was solved iteratively until the relative error
on Res was less than 0.001. Then the vs value was obtained from
Equation 4.2 and the corresponding Cq value was calculated using
Equation 4.3. The advantage of using this second method over the
originally mentioned graphical trial and error method is that it
saves time and minimizes the possible human error in reading Cq
values from the graph. The Cq values obtained by this second meth
od were also checked with the curves presented by Rouse and the
agreement was excellent.
It was already shown on Table 4.2 that the Cp values were in
the range between 15 and 20. That variation is due to the variation
of the kinematic viscosity of the water used on each run. The Cp
value to be used on the evaluation of Equation 3.41 will be the indi
vidual Cq value for each run. No average value of the total runs
will be used on Equation 3.41.
4.2.4 Angle of Repose Evaluation
Kilinc and Richardson (1973) did not present any information on
the angle of repose ($) of the silty sand used in their study. The
angle of repose and the angle of internal friction are intimately
related to each other. Soil characteristics or conditions like the
relative density, dry density, voids ratio, soil strength and
CHAPTER I
INTRODUCTION
1.1 The Soil Erosion Problem
Soil erosion due to precipitation and water flow is a very
complex process. Due to its importance to the agricultural economy
of most societies, it has been studied during the last two or three
centuries, but more seriously so during the last fifty years. It has
been recognized that soil erosion is a natural geological process
shaping the topography of our planet's surface. In addition to caus
ing a general loss of soil to the oceans, erosion reduces the fertil
ity of the soil by carrying away nutrients and minerals that plants
need for a healthy lifecycle; degrades the water quality of natural
or manmade water courses, lakes and oceans; and creates problems to
irrigation, navigation, and water supply systems. Severe soil ero
sion may also result in severe structural problems, including com
plete failure of manmade structures supported by the soil.
The two major soil erosion agents are water and wind (Ellison,
1947). They can erode the soil surface by acting together or indi
vidually. Wind erosion is very important in arid areas and where un
protected soil surfaces (e.g., surfaces not protected by vegetation
or any manmade material) are exposed to the wind. Soil erosion is
also caused by rainfall and water flowing over the soil. Unprotected
soil surface areas are eroded by the rainfall and the subsequent
overland flow carries away the eroded soil particles. The present
1
331
5 10 15 20 25 30 35 40
Bed Slope, S0 (percent)
Figure 5.14. Predicted Probability of Erosion.
137
by regression analysis. Both detachment rate due to rainfall and
critical shear stress concepts were neglected by this simplified
approach.
Some researchers have used stochastic approaches to predict the
sediment yield from overland flow, ephemeral streams (like gullies)
or watersheds. However, the stochastic approach used in the model
depends on the size of the surface area exposed to rainfall and the
area under study (e.g., a plot or a watershed). This happens because
the necessary parameters may not be constant under all conditions
(spatially or temporal variation or their importance among the para
meters may also change).
Studies like Barfield (1968) and Todten (1976) have considered
the sediment transport of suspended particles in shallow flows by
considering the water flow as a two dimensional flow. Each of them
used the diffusion equation under turbulent flow conditions to pre
dict the vertical sediment concentration profile. Unfortunately,
Barfield's model was not able to predict the observed profiles. He
indicated that a possible reason for this was that the model was too
sensitive to the root mean square (RMS) velocity values and any vari
ation of this value was amplified in the diffusion equation. He also
considered that the diffusion coefficients used in his model were too
low because they were calculated from a smooth surface experiment but
used under rough surface conditions.
355
considered. However, the definition of the saltation length is
basically for detached particles which stay in the water while they
are in motion. Splashed grains which travel certain distances
through air cannot be fully accounted for by the proposed defini
tion. However, it is recognized that the number of airborne parti
cles and the distance they travel in air will decrease as the over
land flow depth increases.
Nevertheless, when stuctying the overall soil erosion process,
the proposed model can predict the erosion process by overland flow
with rainfall which is represented by the conditions reported by
Kilinc and Richardson (1973).
6.2 Recommendations
This stucfy has been somewhat limited because of a shortage of
data used to determine the coefficients required in the proposed soil
erosion and transport equation. The equation was developed to esti
mate the soil erosion process of overland flow with rainfall but it
should be useful for a wide variety of flow conditions. Therefore,
future research testing the proposed model at flow conditions not
previously tested may be suggested. It is also necessary to test the
variation of some of the parameters used here (i.e., different types
of soil surfaces with different effective grainsize, varying the
distance along the slope in which data are collected, or considering
the effects of nonuniform slopes).
The suvalue is restricting the general use of the proposed
model because little is known about it and about the raindrop impact
induced turbulence. The remaining parameters, which are needed can
Sh = total head slope
STF = soil properties effect constant for soil transport by
runoff
STT = soil properties effect constant for soil transport by
rainfall
Sy (x xcr)/xcr
Tc = transport capacity of flow
Tc0 = transport capacity of flow at the toe of the sloping bed
Th = total head
t = time
t^ = time consumed for exchange of a particle at the bed
tj = time period of the specific storm increment
t/ig q g= Student's tvalue for 19 degrees of freedom and 5%
v level of significance
U+ /V*
0m = cross section mean velocity of overland flow
USLE = Universal Soil Loss Equation
u = u + u' = local instantaneous longitudinal velocity
u = local timemean longitudinal velocity
u' = local longitudinal velocity fluctuation
u^ = lateral velocity of water moving away from waterdrop impact
area
umav = maximum local timemean velocity at distance ymav
n,ax from the bed surface max
u^ = Â¡Â¡Â£ + ui = instantaneous velocity near top of grains on the
bed
ut = timemean velocity near top of grains on the bed
xx ii i
24
where KEt has units of J/(ha hr), I is in mn/hr and Cte is a tem
perature correction factor.
Rogers et al. (1967) discussed some of the sources of error in
calculating the kinetic energy of rainfall. They indicated that the
sources of errors are variations in the raindrop size distribution
even at different periods of the rainstorm with the same rainfall
intensity and the measuring technique used to measure rainfall inten
sity and wind effects.
Recently, Mualem and Assouline (1986) proposed an analytical
function to represent the raindrop size distribution which was cali
brated for Rhodesia (Hudson, 1965) and Washington, D. C. (Laws and
Parson, 1943) data. From it, the rainfall kinetic energy per unit
mass and the rainfall kinetic energy per unit time expressions were
presented as a function of rainfall intensity. The curves for rain
fall kinetic energy per unit mass differed significantly from known
empirical expressions obtained by other authors which used the same
data. Their rainfall kinetic energy per unit time curve was found to
have an insignificant deviation between both data places at low rain
fall intensities, but became noticeable at higher values of the rain
fall intensity.
Similarly, there are some relationships giving the momentum of
rainfall applied to a given surface and the rainfall intensity men
tioned in the literature. Elwell and Stocking (1973) used the
expression originally developed in Hudson's masters thesis (1965)
130
governing equations they derived closed form equations for detachment
rate and sediment load along the sloped bed.
D* = (1 9*) (1 e"***) (2.73)
G* = x*
(l ej
(1 e
cux
*A*
) = x* 
Dco
(2.74)
where
D* = flow detachment rate relative to flow transport capacity
at the toe of the slope = (L0Dp/TC0)
G* = sediment load relative to flow transport capacity at the
toe of the slope = (gs/TC0)
x* = distance from the top of the slope relative to the total
length of slope (x/L0)
0* = rainfall detachment parameter = (L0 DIo/Tco>
a* = runoff detachment parameter = (L0 Dc0/Tc0)
L0 = total length of the uniform sloped bed
and the parameters with subscript o representing the parameter values
at the toe of the sloped bed.
For deposition conditions they indicated that DÂ¡ and thus 0*
were zero which gave the equations for deposition and sediment load
an exponential decay as the distance along the slope increased.
Using this approach, Foster and Meyer were able to predict deposition
as the bed slope flattened (and the flow lost its transport capacity)
without using a gradually varied flow analysis.
Later, Foster and Meyer (1975) used this approach for cases of
nonuniform slope (concave or convex slopes) with net erosion or net
187
studies, nQ will also be assumed equal to 3.09 for t0 = Tcrs. In
this form both studies will have the same erosion equation for open
channel flows without rain on horizontal or nearly horizontal beds.
It should also be mentioned that Chiu's definition of B2 is the
? 1/2
B2 definition used in this study multiplied by (l/su)(l + suc)
From both methods the erosion criterion is the same expression when
the corresponding values of &2 are used.
3.3.2 Probability of Erosion
As mentioned before, rainfall increases the probability of ero
sion by increasing the bed shear stress and the turbulence intensity
of the flow. By assuming that the velocity fluctuations are Gaussian
distributed, the probability of erosion can still be obtained using
the Gaussian distribution. Using the n values given by Equations
3.37 and 3.38 as the limits of the integration of the probability
density function, the probability of erosion may be written (see Fig
ure 3.7) as
p = Area} + Area2
P =
1
e <"2/2) dn
1
/2tt
e dn
(3.39)
n+
Chiu showed that using the values of n0 = 3.09 and a value of
su = 0.164 obtained analytically by Christensen (1965), the Area^
below the probability curve must be very small compared to Area2
284
a better idea of the standard deviation of each of the estimated co
efficient values and the corresponding confidence limit of the re
sults. The variance of each variable of every data point is unknown.
The estimated relative errors cannot be used for this purpose because
they are only estimates based on a constant relative error for every
data point of any variable (Table 5.1). In this case, a weighing
factor of one is probably the best alternative to use. This means
that the error of each data point of a given variable may be the same
as the other data point errors of this variable.
This consideration does not affect the value of coefficients
obtained by the least squares method, but does affect the standard
deviation of the estimated values. Brandt (1976) considered this
condition and explained it (Example 122, pages 284289) using dif
ferent (unknown and known) values of the error of the data. For the
case in which the error of the data is unknown, Brandt indicates that
the model can only be supported by visual inspection of the predicted
results and the observed values. Based on the results shown in Fig
ure 4.10, the model apparently is good enough to represent the trends
shown by the data. More discussions about the validity of the model
are presented in the following sections.
5.3 Error Analysis of the Model and the Predicted Values
Equation 3.41 is the basic equation of the model. It is also
accompanied by the definitions for C* and su given by Equation 4.40
and Equation 4.41, respectively. Based on Kilinc and Richardson's
(1973) data, the value of the coefficients of those equations were
SOIL EROSION BY
OVERLAND FLOW WITH RAINFALL
BY
RAUL EMILIO ZAPATA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
Copyright 1987
by
Raul Emilio Zapata
To Carmencita,
Raul Enrique and Mari Luz
ACKNOWLEDGMENTS
I want to express n\y most sincere appreciation to Dr. Bent A.
Christensen, committee chairman, for the direction, advice and assis
tance which he has given to me throughout my graduate studies at the
University of Florida. His knowledge, moral support and patient
guidance helped me to complete this study.
Thanks are due to Dr. E. R. Lindgren for his teaching lessons
in fluid mechanics and for serving on the supervisory committee.
Thanks are also extended to Dr. L. H. Motz for serving on the super
visory committee.
Special thanks are extended to the University of Puerto Rico,
Mayaguez Campus, for providing me the opportunity to improve my know
ledge, securing me a leave of absence and financial support through
out try studies. Thanks are also extended to the Center for Instruc
tional Research Computing Activities at the University of Florida for
the use of their facilities.
Thanks are extended to Gail Luparello and Irma Smith for the
quality typing, and also to Katarzyna Piercey and her husband, Dr. R.
Piercey, for their beautiful drawings.
To ny wife, Maria del Carmen, whose loving support and encour
agement has always inspired me and allowed me to have beautiful
iv
experiences with our children, Raul Enrique and Mari Luz, I give my
deepest love, appreciation, and respect. I am also very grateful to
n\y parents and family for their patience and understanding during
this period of our lives.
I wish to express my deep appreciation to Dr. L. Martin, Dr. F.
Fagundo and their families for the friendship, guidance and help
which they have provided to me and my family during our stay in this
natural and beautiful city of Gainesville, Florida. I thank my fel
low graduate students and neighbors for their friendship and encour
age them to continue working hard to reach their goals.
v
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iv
LIST OF TABLES x
LIST OF FIGURES xii
LIST OF SYMBOLS xv
ABSTRACT xxviii
CHAPTERS
I. INTRODUCTION 1
1.1 The Soil Erosion Problem 1
1.2 Purpose and Scope of This Study 5
II. SOIL EROSION PROCESS AND REVIEW OF
RELATED STUDIES 7
2.1 The Soil Erosion Process 7
2.2 Initial Studies 11
2.3 Raindrop and Rainfall Characteristics .... 15
2.3.1 Raindrop Characteristics 15
2.3.2 Rainfall Characteristics 19
2.4 Splash Erosion 25
2.4.1 Waterdrop Splash 25
2.4.2 Splash Erosion Studies 32
2.5 Overland Flow Erosion 39
2.5.1 Hydraulics of Overland Flow 39
2.5.1.1 Simplified Solutions, the
Kinematic Wave Method .... 51
2.5.1.2 The Law of Resistance .... 55
2.5.1.3 Boundary Shear Stress .... 62
Kinematic Approach 62
Dynamic Approach 63
2.5.1.4 Entrainment Motion and
Critical Shear Stress .... 66
2.5.1.5 Flow Velocity 77
2.5.1.6 Turbulence 87
2.5.2 Overland Flow Erosion Studies 93
vi
2.6 Rill and Gully Erosion 100
2.6.1 Rill Erosion 100
2.6.2 Gully Erosion 102
2.7 Soil Erosion Estimates and Prediction .... 103
2.7.1 Use of Existing Stream Sediment Trans
port Equations in Overland Flow . . 104
2.7.2 The Universal Soil Loss Equation ... 114
2.7.3 Soil Erosion Models 123
2.8 Soil Characteristics and Slope Effects in
Soil Erosion 144
2.8.1 Soil Characteristics 144
2.8.2 Slope Gradient Effects 148
2.9 Recent Books on Soil Erosion 151
III. DEVELOPMENT OF THE SOIL EROSION EQUATION 153
3.1 General Purpose and Considerations 153
3.1.1 Basic Considerations and Assumptions 153
3.1.2 Major Considerations and Assumptions 154
3.2 Equilibrium Transport Condition 156
3.2.1 Evaluation of Ne 157
3.2.2 Evaluation of Nd 159
3.2.3 Evaluation of 159
3.2.4 Evaluation of Average Saltation
Length, i 165
3.2.5 General Equilibrium Transport Equation. 166
3.3 Relationships between Probability of Erosion
and Bed Shear Stress 169
3.3.1 Criterion for Erosion 169
3.3.2 Probability of Erosion 187
3.4 Sediment Transport Equation 189
IV. DATA, PROCEDURES, AND COEFFICIENTS EVALUATION ... 191
4.1 Introduction 191
4.2 Data 191
4.2.1 Soil Properties 192
4.2.2 Effective Grain Size Evaluation .... 199
4.2.3 Drag Coefficient Evaluation 200
4.2.4 Angle of Repose Evaluation 202
4.2.5 Shields Entrainment Function and
Critical Shear Stress Evaluations . 203
4.2.6 TimeMean Bed Shear Stress Evaluation 204
4.2.7 Grain Reynolds Number Evaluation . 204
4.2.8 Dimensionless Sediment Transport
Parameter 204
4.2.8.1 Evaluation of $ 204
4.2.8.2 Evaluation of'i'' and 4' . 205
4.2.9 Sediment Transport Data in Diagrams . 206
vii
4.3Evaluation of C+ and su 214
4.3.1 Additional Considerations on the
C^Value 214
4.3.2 su Considerations 239
4.3.3 Procedure Used to Evaluate the
Coefficients 247
4.3.4 The Values of the Coefficients C2, C4,
Cg, Cg, and m 250
V. DISCUSSIONS AND MODEL VERIFICATION 255
5.1 Introduction 255
5.2 Error Analysis of the Data 255
5.2.1 Estimated Relative Error of q .... 257
5.2.2 Estimated Relative Error of 259
5.2.3 Estimated Relative Error of 'F' 259
5.2.4 Estimated Relative Error of the Slope
Correction Factor of 'Fq 259
5.2.5 Estimated Relative Error of 'i' .... 260
5.2.6 Discussion of the Error of the
Longitudinal Slope Correction Factor 261
5.2.7 Discussion of the Estimated Relative
Errors of $ and V1 268
5.2.8 Other Possible Errors 281
5.2.9 Use of the Estimated Data Errors in
Evaluation of the Coefficients .... 283
5.3 Error Analysis of the Model and the
Predicted Values 284
5.3.1 General Statistics of the Model .... 285
5.3.2 Statistical Analysis of the Estimated
Coefficient Values 288
5.3.3 Discussion of Errors of the Model
Predicted Values 296
5.3.4 Justification of the Least Squares
Approximation Method 307
5.4 The Saltation Length Process and the
C Values 313
5.5 The suValues 327
5.6 Final Remarks 345
VI. CONCLUSIONS AND RECOMMENDATIONS 352
6.1 Conclusions 352
6.2 Recommendations 355
APPENDICES 359
A GENERAL NOTES IN THE EVALUATION OF THE ABSOLUTE
ERROR AND RELATIVE ERROR OF VARIABLES 359
viii
B INFLUENCE OF CHANGE OF DIRECTION OF BUOYANT FORCE
ON PROPOSED MODEL 361
C CONVERSION FACTORS 371
REFERENCES 373
BIOGRAPHICAL SKETCH 400
ix
LIST OF TABLES
Table Page
2.1 RANGE OF VALUES OF VARIABLES AND SOME DIMENSIONLESS
PARAMETERS IN OVERLAND FLOW 46
2.2 RELATIVE MAGNITUDES OF THE TERMS S' Si, So and So in
TERMS OF Sf 48
2.3 SOME STREAM BEDLOAD TRANSPORT FORMULAS 105
4.1 RELEVANT DATA OBTAINED FROM KILINC AND RICHARDSON'S STUDY. 193
4.2 DIMENSIONLESS PARAMETERS CALCULATED FROM KILINC AND
RICHARDSON'S DATA 195
4.3 CRITICAL SHEAR STRESS AND DIMENSIONLESS SEDIMENT
TRANSPORT VARIABLES 197
4.4 IDENTIFICATION OF DATA POINTS AND GENERAL LEGEND
FOR FIGURES IN THIS STUDY 211
5.1 ESTIMATED RELATIVE ERROR OF THE DATA 256
5.2 SEDIMENT CONCENTRATION VALUES AND THEIR CALCULATED
RELATIVE ERROR 258
5.3 ESTIMATED ERROR OF THE LONGITUDINAL SLOPE CORRECTION
FACTOR 262
5.4 ESTIMATED RELATIVE ERROR OF $, yJf AND RELATED VARIABLES 264
5.5 DATA POINTS WITH POTENTIALLY LARGE ERRORS OF y' 270
0
5.6 AVERAGE RELATIVE ERROR OF $ AND Cs FOR EACH DATA SET
WITH SAME RAINFALL INTENSITY 279
5.7 AVERAGE RELATIVE ERROR OF $ AND Cs FOR EACH DATA SET
WITH SAME BED SLOPE 279
5.8 DATA POINTS WITH POSSIBLE LARGE ERRORS OF $ 282
5.9 ANALYSIS OF VARIANCE 286
5.10 STATISTICAL ANALYSIS OF THE COEFFICIENTS IN THE MODEL . 289
x
5.11 REQUIRED MINIMUM WATER DEPTH TO USE THE PROPOSED MODEL . 293
5.12 DATA POINTS WITH PREDICTED ERRORS LARGER THAN THE
ESTIMATED ERROR OF DATA 300
5.13 PREDICTED $VALUES AND THEIR ERRORS 305
5.14 SUMMARY OF DATA POINTS WITH LARGEST ERROR ON THE
PREDICTED MODEL SOLUTION 306
5.15 RAINFALL INTENSITY EFFECTS IN f(h/de) FUNCTION FOR
GIVEN WATER DEPTH 318
5.16 PREDICTED su AND pVALUES 333
B.l ANALYSIS OF VARIANCE USING IMPROVED METHOD (EQUATION
3.22) 365
B.2 STATISTICAL ANALYSIS OF THE COEFFICIENTS IN THE MODEL . 366
B.3 PREDICTED $VALUES AND THEIR ERRORS 367
B.4 PREDICTED su AND pVALUES OF THE IMPROVED METHOD 368
xi
LIST OF FIGURES
Figure Page
2.1 Definition Sketch 44
2.2 ErosionDeposition Criteria for Uniform Particles .... 68
2.3 Shields' Diagram for Incipient Motion Including Mantz
Extended Curve for Fine Cohesionless Grains 70
2.4 Typical Velocity Profile and Shear Stress Distributions
for Flow With and Without Rainfall 82
2.5 Surfaces of Detachment, Transport and Maximum
Erosion Rate 126
3.1 Particle Travel Distance 160
3.2 Schematic Saltation Length Approach for Overland Flow . 163
3.3 Effect of Longitudinal Slope on the Saltation Length
of a Grain 167
3.4 Forces Acting on a Grain About to Move for the Horizontal
Bed and Sloping Bed Conditions 171
3.5 Incipient Motion for Uniform Flow Condition Following
Ulrich's (1987) Approach 174
3.6 Correction Factor on the Bed Shear Stress due to
Longitudinal Slope for Ulrich's Coarse Material 176
3.7 Evaluation of the Probability of Erosion 188
4.1 GrainSize Distribution of the Sancfy Soil Used by
Kilinc and Richardson (1973) 201
4.2 $ Versus t': The Data Points 207
4.3 $ Versus T' for Given Rainfall Intensity 208
4.4 $ Versus V^ for Given Rainfall Intensity 209
4.5 Versus 'F for Given Bed Slope 210
xii
4.6 Initial Depth Required to Move Grain Under Very
Shallow Water Depth Conditions 220
4.7 Slope Effects on the Normal Component of the
Rainfall Momentum Flux 230
4.8 Predicted suValue Using C* = 2.2 x 107 ft^ (Constant) 245
4.9 Predicted suValue Versus Measured Sediment
Concentration using C* = 2.2 x 107 ft (Constant) . 246
4.10 Comparison of Observed and Predicted Dimensionless
Sediment Transport 252
4.11 Predicted and Required suValues 253
5.1 Bed Shear Stress Ratio 267
5.2 To_yaues as Calculated by Kilinc and Richardson (1973) 272
5.3 ^'Values for Given Bed Slope 273
5.4 ^'Values for Given Bed Slope 274
0
5.5 Observed ^Values for Given Bed Slope 277
5.6 Predicted $> and Estimated Error Ranges in Data for
Rainfall Intensities of 2.25 and 4.60 in./hr 297
5.7 Predicted $ and Estimated Error Ranges in Data for
Rainfall Intensities of 1.25 and 3.65 in./hr 298
5.8 Predicted $ Versus Observed $Values 303
5.9 Residual Values Versus the Natural Logarithm of
Observed i>Values 310
5.10 Normal Probability Plot of the Standardized Residual . 311
5.11 Saltation Length Depth Function, f(h/d0) 316
5.12 Slope Correction Factor for the Average Saltation Length. 320
5.13 Saltation Length Ratio, 2.2 x 107/C* 324
5.14 Predicted Probability of Erosion 331
5.15 Changes in p/(l p) Due to Errors in p Evaluation . 336
xi ii
5.16 Required suValue Versus Measured Sediment
Concentration, C$ 341
5.17 Predicted suValues Versus Measured Sediment
Concentration, C$ 342
5.18 Relationship Between C$ and v+dg/v 344
B.l Correction Factor for Bed Shear Stress due to Longi
tudinal Slope for Kilinc and Richardson's Silty
Sand Material 363
B.2 Required suValues 370
xiv
LIST OF SYMBOLS
A = cross sectional area of water flow
A0 = surface area exposed to falling raindrops
A} = constant of particle area
A2 = constant of particle volume
A3 = A3 (/(2A2) = constant
A3 = dimensionless constant
AÂ¡ = increment of surface area
a = 2.5
a0 = coefficient between 0 and 1 used by Onstad et al. (1976)
aj = constant
a2 = constant
a^ = thickness of the bedload transport layer, assume twice the
size of sediment particles
ad = kl = constant in DuBoys formula
af = coefficient relating detachment capacity to transport
capacity of flow
aÂ¡ = coefficient which depends in soil characteristics
a^ = coefficient used in discharge per unit width equation of the
kinematic wave method
am = empirical coefficient
ap = constant in velocity profile equation
ar = coefficient to relate rainfall intensity to the roughness
coefficient, K
ay = 2.45 x(y/ys)0,4
xv
B =
width of the crosssectional area of the flow
B0 =
buoyant force of a particle in a static fluid (horizontal
water surface)
b =
7.0
bk =
coefficient used in discharge per unit width equation of the
kinematic wave method
bm =
2.1 Clf
II
Q
Q
constant in velocity profile equation
br =
coefficient to relate rainfall intensity to the roughness
coefficient, K
C =
Chezy's coefficient
^0 =
constant determined by Chiu for deep water flow conditions
Cl =
p
dimensional function for the saltation length (length ~c)
c2 
constant representing initial dimensionless water depth
required to have incipient grain motion on a horizontal bed
C3 =
dimensionless constant
c4 =
dimensionless constant related to rainfall intensity
influence in water depth function of the saltation length
definition
c5 =
suvalue when v+dg/v = 1
o
cr>
ii
(1/2.3)*(slope of the su versus Â£n(v*de/v) curve)
ca =
sediment concentration near the top of the bed layer
Cc
canopy density cover factor
CD =
drag coefficient
cg =
ground density cover factor
cif =
clay fraction percent
Cm =
cropping management factor in USLE
Cmi
cropping management factor for intern'1 area
xvi
Cmr = cropping management factor for rill area
Cs = = total sediment concentration in the water flow
Ctp = temperature correction factor in energy equation, Park et
al. (1983)
Ctm = temperature correction factor in momentum equation, Park
et al. (1983)
C = dimensionless friction coefficient
T
c* = a1a3/a2c1
Cf = DarcyWeisback friction factor
c1 = 8g Sf0,1/(1.481,8 *2)
c" = c'Nm1,8
D = Drag Coefficient
D50 = mean equivalent spherical raindropsize diameter for given
rainfall intensity
Dc = detachment capacity of flow
Dc0 = detachment capacity of flow at the toe of the sloping bed
De = equivalent spherical raindropsize diameter
Dp = soil detachment by runoff
DÂ¡ = soil detachment by rainfall
D' = ~V*S 1
du
D* *0df/tco
d = ds = grainsize diameter
d1Q = grain size with 10% of finer material
d31 = 9raln size w1th 31^ of: finer material
d35 = 9ran size with 35% of finer material
d50 = 9rain S1ze w1th 50^ f finer material
xv ii
d54 = 9rain size w1th 54^ of ^iner material
dgy = grain size with 57% of finer material
dÂ£ = de value used in this study = 145 pm = 4.76 x 10~4 ft
de = diameter of effective grain size
E = rate of soil loss from USLE
Eh = critical value of Shields' entrainment function
Ej = soil erosion on intern'll areas
EV = estimated coefficient value
e = base of natural logarithm
en = void ratio of the soil
F = fraction of weight of the sediment that is finer than grain
size d
Fq = resultant detachment force
Fdh = resultant detachment force for horizontal bed
Fqc; = resultant detachment force for sloping bed
Fr = resultant force at incipient motion
Fr = 0m/(gh)*/2 = Froude Number
Ftest value = statistical value used to test hypothesis
f = infiltration rate
f(en) = 0.685/(1 + en)0,415
f(h/de) = function to represent the water depth influence in the
saltation length
f(I) = function to represent the rainfall properties in the water
depth function, f(h/de)
f(IVt) = function to represent the rainfall parameter effects in
the saltation length
xv ii i
f(e) = function to represent the longitudinal slope influence in the
saltation length
G = weight of grain in air at one atmosphere of pressure
G+ = sediment load relative to flow transport capacity at the
toe of the sloping bed
g = acceleration of gravity
gs = sediment load (weight per unit time per unit width)
gse = total soil loss mass per unit width in a storm event
h = water depth, measured normal to bed surface
h^ = required water depth to have incipient grain motion on
horizontal beds
hÂ¡ = initial depth required to reach incipient motion
hm = average water flow depth
hs = h^ cos 6
hw = water depth plus loose soil depth
h1 = local water depth at distance x' from the bank
h* = overpressure head induced by the raindrop impacts over the
hydrostatic pressure head.
I = rainfall intensity
130 = rainfall intensity during the maximum measured 30minute
rainfall intensity during the rainstorm event
I* = I f = rainfall excess
K = roughness coefficient associated to Cf and Ref
KEa = rainfall kinetic energy per unit area
KEq = waterdrop kinetic energy
KEf = rainfall kinetic energy per unit area per unit time
Kf = soil erosivity factor in USLE
xix
KfC = soil erosivity factor
Kfe = soil erosivity factor for channel erosion
K0 = Kvalue with no rainfall conditions
k = roughness size of the bed surface
L = lift force
Le = overland flow length
Lf = slope length factor in USLE
L0 = total length of the sloping bed
Â£ = average saltation length
Â£* = average total distance traveled by a particle before it is
finally at rest
= average saltation length on horizontal bed
in = natural logarithm
Â£og = logarithm to base 10
Â£5 = average saltation length on sloping bed
= rainfall momentum per unit area
Ml = exponential coefficient based on the bed slope and used in USLE
Mt = rainfall momentum per unit area per unit time
MUSLE = Modified Universal Soil Loss Equation by Williams (1975)
m = dimensionless exponential coefficient used in f(e )
N = number of particles in motion
ND = number of data points
N(j = number of particles deposited per unit time and unit bed area
Ndrop = number of raindrops collected on a given area per unit time
Ne = number of particles eroded per unit time and unit bed area
xx
= Manning's roughness coefficient
NMb = Manning's roughness coefficient for bare soil
NMc = Manning's roughness coefficient for rough, mulch or
vegetative covered soil
n = normalized velocity fluctuation
n0 = 3.09 = value of n corresponding to = Tcrs
n = limit of integration to obtain probability of erosion from
+ Area
n = limit of integration to obtain probability of erosion from
Area^
ns = number of straight lines into which the grainsize distribution
curve is divided
OMF = overland momentum flux
P = pressure
P* = overpressure due to raindrop impacts
Pf = erosion control practice in USLE
P^ = erosion control practice for intern'll areas
Pfr = erosion control practice for rill areas
p = absolute probability that a particle is eroded
Q = water flow discharge
q = water discharge per unit width
qÂ¡_ = lateral flow discharge per longitudinal unit length
qp = storm runoff peak
qs = volume of particles with size de transported per unit time
and unit bed width
R = Rei = rainfall erosivity factor in USLE
R' = hydraulic radius
R = resistance radius
xx i
ReA = vDDe/vA
Rede = v*Vv
Ref = Om/h/v
Res = vsde/v
RMF = rainfall momentum flux
RMFn = rainfall momentum flux normal to bed surface
r = correlation coefficient
r^ = coefficient of determination
S = slope gradient factor in USLE
50 = tan 0
S = sine
51 = l (i M2)
3 x gh
2glq
gh2
I VD cosfl
gh
s3 =
SC(
scu =
v s tan e
>DF
>DI
SEE =
sf =
SGs =
y$ y tan $
soil properties effect constant for soil detachment by
runoff
soil properties effect constant for soil detachment by
rainfall
slope of energy grade line
standard error of estimate
friction slope
specific gravity of particle is fluid in water
xxi i
Sh = total head slope
STF = soil properties effect constant for soil transport by
runoff
STT = soil properties effect constant for soil transport by
rainfall
Sy (x xcr)/xcr
Tc = transport capacity of flow
Tc0 = transport capacity of flow at the toe of the sloping bed
Th = total head
t = time
t^ = time consumed for exchange of a particle at the bed
tj = time period of the specific storm increment
t/ig q g= Student's tvalue for 19 degrees of freedom and 5%
v level of significance
U+ /V*
0m = cross section mean velocity of overland flow
USLE = Universal Soil Loss Equation
u = u + u' = local instantaneous longitudinal velocity
u = local timemean longitudinal velocity
u' = local longitudinal velocity fluctuation
u^ = lateral velocity of water moving away from waterdrop impact
area
umav = maximum local timemean velocity at distance ymav
n,ax from the bed surface max
u^ = Â¡Â¡Â£ + ui = instantaneous velocity near top of grains on the
bed
ut = timemean velocity near top of grains on the bed
xx ii i
u = velocity fluctuation near top of grains on the bed
Vp = waterdrop velocity
Vj = terminal velocity of raindrop with equivalent spherical
raindrop size Dj
Vr = storm runoff volume
Vt = mean terminal velocity of the raindrops
v = v + v' = instantaneous vertical (normal) velocity at a given
location
v = timemean vertical velocity
v1 = vertical velocity fluctuation at a given location
v* = shear velocity
v*cr = c^tieal shear velocity
v*s =
vs = particle fall velocity
vs = fall velocity of particle with size d35
3 3
W = buoyant weight of grain
We = p h 0m2/ r
WS = weight of splashed soil by single waterdrop impact
w = w + w' = instantaneous lateral velocity at a given location
w = timemean lateral velocity
w' = lateral velocity fluctuation
X_ = slope length
x = longitudinal distance
x' = distance measured across the flow from its bank
x* = x/Lo
xcr = critical length to initiate erosion
xx iv
Y+ = yv^/v
y distance from the bed surface to a location in the water
ymax = distance,from the bed surface to the location with
maximum u
1\\ = depression storage elevation on a horizontal bed
z = vertical distance from the bottom surface
a = dimensionless energy correction factor for the velocity
distribution of the flow
ai = level of significance for tStudent test
B
Bl
B2
Bl
Bl
r
Y
Yd
Ys
AX
6
0
= dimensionless momentum flux correction factor for the velocity
distribution of the flow
= constant of particle area
= (1 + n0su)/(1 + su2)1/2
= dimensionless momentum flux correction factor for the
distribution of the raindrop terminal velocity
= dimensionless momentum flux correction factor for the lateral
flow velocity distribution
= surface tension of water
= specific weight of water
= specific dry weight of soil material including pore volume
= specific weight of soil grains
= longitudinal length increment
= thickness of the viscous sublayer
= very small number compared to unity
= very small value of SCU
= longitudinal bed surface inclination with respect to the
horizontal
xxv
H I
e* *o Dio/Tco
K = von Krman constant
X = instantaneous lift per unit area
Xj = constant for saltation length
p = dynamic viscosity of water
v = kinematic viscosity of water
VA = kinematic viscosity of air
Â£ = water surface angle with respect to the horizontal
p = mass density of water
ogrr = standard deviation of the estimated error
1 = soil shear strength
T0 = instantaneous bed shear stress
T0 = timemean bed shear stress
tcr = timemean bed shear stress when p = 10"3
Tcrs = timemean bed shear stress for sloping beds when p
Tf = timemean shear stress due to form roughness
g = timemean shear stress due to grain roughness
s = timemean shear stress at the water surface
$ = sediment transport intensity function
<(> = angle of repose
xx vi
X
= V((Ys Y)ds) = 1/T'
Ycr Tcr^ ^Ys Y)ds)
t' = flow intensity function
Â¥q = flow intensity function for sloping bed surfaces
Si = angle of the path of the falling raindrops with respect to the
vertical axis
w = angle of detachment of the resultant force with respect to the
bed surface
H = detachment angle for horizontal bed
>5 = detachment angle for sloping bed
xxvi i
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SOIL EROSION BY
OVERLAND FLOW WITH RAINFALL
By
Raul Emilio Zapata
December 1987
Chairman: Dr. Bent A. Christensen
Major Department: Civil Engineering
The objective of this study is to develop a soil erosion model
for overland flow with rainfall based on physical concepts and obser
vations. The basic sediment transport equation used by the proposed
model is based on the modified Einstein equation for total load
transport of noncohesive materials in open channels as presented by
Chiu in 1972. The proposed model was developed as generally as pos
sible in order to be valid for the case of deep water flows (i.e.,
rivers and open channels) as well as for very shallow flows (overland
sheet flow) with or without rainfall.
The proposed model provides a stochastic point of view of this
random process, usually modeled using deterministic approaches.
Rainfall effects on the erosion process are mostly represented in the
changes of the boundary shear stress and in the local velocity fluc
tuation at the top of the grains. The timemean bed shear stress is
obtained from the longitudinal momentum equation including additional
xxviii
momentum terms due to the incoming rainfall flux. The local velocity
fluctuations are assumed to be distributed according to the Gaussian
law.
Chiu's saltation process assumed the saltation length to be
inversely proportional to the particle size and independent of the
flow conditions. In this study the effect of the longitudinal bed
slope, water depth, and rainfall parameters on the saltation length
is included. However, distance traveled in air by particles splashed
by raindrops and the number of particles which travel airborne cannot
be fully accounted for by the proposed definition.
An error analysis was conducted to the proposed model and the
assumptions made during its development. Errors on the predicted
values were found to be similar to data errors. The maximum relative
error on the predicted values was 33%, which may be considered rea
sonable for the complex process of soil erosion by overland flow with
rainfall. The proposed model may be used in the future as an initial
step toward an improved erosion model based on the physics of this
complex process.
xx ix
CHAPTER I
INTRODUCTION
1.1 The Soil Erosion Problem
Soil erosion due to precipitation and water flow is a very
complex process. Due to its importance to the agricultural economy
of most societies, it has been studied during the last two or three
centuries, but more seriously so during the last fifty years. It has
been recognized that soil erosion is a natural geological process
shaping the topography of our planet's surface. In addition to caus
ing a general loss of soil to the oceans, erosion reduces the fertil
ity of the soil by carrying away nutrients and minerals that plants
need for a healthy lifecycle; degrades the water quality of natural
or manmade water courses, lakes and oceans; and creates problems to
irrigation, navigation, and water supply systems. Severe soil ero
sion may also result in severe structural problems, including com
plete failure of manmade structures supported by the soil.
The two major soil erosion agents are water and wind (Ellison,
1947). They can erode the soil surface by acting together or indi
vidually. Wind erosion is very important in arid areas and where un
protected soil surfaces (e.g., surfaces not protected by vegetation
or any manmade material) are exposed to the wind. Soil erosion is
also caused by rainfall and water flowing over the soil. Unprotected
soil surface areas are eroded by the rainfall and the subsequent
overland flow carries away the eroded soil particles. The present
1
2
stucty is only concerned with the soil erosion due to rainfall impact
and flowing water. Most of the observations presented here are based
on water. Nevertheless, wind erosion is recognized as an eroding
agent which has a significant effect in the shaping of the earth's
surface.
The soil erosion problem has been widely studied by many scien
tists and engineers, but it is still not well understood. The main
reason is that soil erosion is a very complex process and involves so
many variables that it is practically impossible to measure the
influence of all of them in one study.
In addition, the scale of this process is so small that it
makes it practically impossible to measure the variables in an accur
ate manner even with today's sophisticated equipment. Researchers
cannot produce a model with dimensions longer than those of the pro
totype because not all of the dimensions or variables can be modeled
to the same scale (e.g., waterdrops larger than 7 mm are unstable
(Blanchard, 1950) and the terminal velocity of the waterdrops is
influenced by surface tension while in a larger model surface tension
may be relatively negligible). Using the prototype size requires
equipment with almost microscopic dimensions in order to measure flow
parameters at different locations rather than average values with
larger instruments. Therefore, scaling and instrumentation are
problems that soil erosion researchers have to deal with.
Soil erosion is also an unsteady and stochastic process. Des
cribing it requires knowledge of how the variables change with time.
In most of the studies, the investigators have selected some
3
significant variables or parameters and studied their behavior under
different conditions. Unfortunately, most of these studies were con
ducted under simulated steady or quasisteady conditions. This is
considering that the surface area under study is significantly larger
than the area affected by a single raindrop impact and that the time
period is substantially larger than the time increment between rain
drop impacts. Under such conditions, the soil erosion process can be
mathematically described as a simpler process, but the capability to
predict changes in time and sometimes spatial changes is lost.
There are other studies in which a specific part of the soil
erosion process is considered and measured. Then a mathematical
model is developed and used in describing that part of the erosion
process.
Some of the general topics related to soil erosion studied in
the past are
 Relation of rainfall to runoff and soil erosion.
 Detachment and/or transportation of soil particles by
raindrop impact.
 Detachment and/or transportation of soil particles by
runoff.
 Soil erosion caused by the combined actions of rainfall and
runoff.
 The erosion process related to hydraulic parameters.
 The influence of soil properties on the erosion process.
 The influence of longitudinal slope, slope length and/or
slope shape on erosion.
 Measurement and prediction of soil loss from a given area.
 Eroded soil characteristics and properties.
 Nutrient and pollutant migration, etc.
4
Most of the studies found in the literature seem to describe
empirical approaches where, from observed data, equations are devel
oped using some kind of a regression analysis. One disadvantage of
this approach is that the developed equations are only valid for the
specific conditions that existed during the observations. Extrapola
tion, of course, cannot be recommended, and even interpolation may
have its problems. The other disadvantage of using empirical ap
proaches is that the resulting empirical relationships can only pre
dict the mean values of the observations for a certain condition and
any information relating to fluctuations is lost. Some of the varia
bles upon which many of these studies are based are the kinetic ener
gy or momentum of raindrops, the rainfall intensity, some of the soil
properties, vegetative cover, slope and length of the ground surface
and the conservation practice used in order to prevent soil loss.
Attempts have been made to explain soil erosion by the basic
physical laws. However, the results are usually limited to very nar
row parts of the whole process. Continued research is definitely
needed. Attempts to physically describe the effect of raindrops and
sheet flow at the same time have been only partially successful and
the literature on this topic is quite limited.
Attempts of using existing sediment transport equations origin
ally developed for water courses have been made, but the results have
not been satisfactory because the boundary conditions are different.
5
Modification of such equations in order to consider the effect of
very thin surface flows and rainfall may however improve prediction.
However, caution is advised in order to satisfy all boundary condi
tions at any time and location. Real physical models describing the
soil erosion process in overland flow are very rare. This study
attempts to present a new vision of this erosion process which might
help others in the future to better understand this very complex
process.
1.2 Purpose and Scope of This Study
The goal of the present investigation is to describe the soil
erosion process using a stochastic approach similar to that original
ly presented by Chiu (1972) for transport of cohesionless sediments
by water and air. That approach is based on the saltation hypothesis
and may be considered as a modification of the original stochastic
theory presented by Einstein (1950).
To reach that goal only physical considerations will be used.
The proposed approach is intended to be as simple and general as pos
sible. So it will allow use of the model at almost any flow condi
tion if the required information is available. However, before use,
it will be necessary to test the proposed approach at these not pre
viously tested flow conditions, even though the required basic physi
cal considerations are expected to be included into the proposed
model of this investigation.
The use of the stochastic approach requires some modifications
in order to include the rainfall effect on the thin sheet flow.
6
Those modifications will be presented in Chapter III and IV where the
proposed approach is presented and developed using existing data from
the literature. In addition, an error analysis of the used data and
the predicted results is presented and discussed in Chapter V. The
possible advantages and limitations of the proposed method are
discussed. Other discussions and conclusions are also presented.
Throughout this work the unit system used is the English Sys
tem. However, in the review of other studies related to soil erosion
the unit systems used in those studies are used. The proper unit
conversion is presented in such cases. See also Appendix C for the
conversion factors for units between the English System and the SI
System.
CHAPTER II
SOIL EROSION PROCESS AND REVIEW OF RELATED STUDIES
2.1 The Soil Erosion Process
The initial cause of soil erosion due to water is rainfall.
When the raindrops impact a ground surface not covered by water,
their kinetic energy will generate a splash of water in which thou
sands of droplets will disperse in all directions (Mutchler and Lar
son, 1971; and Mutchler, 1971). Some of these droplets will carry
soil particles out of the area of impact. The amount of soil de
tached and the distance traveled by the individual soil particles
will be a function of the ground surface soil properties and the
rainfall characteristics.
If there is a water layer covering the ground surface, the ef
fect of the splash on the soil surface will be mostly a function of
the water layer thickness, the drop diameter and the soil properties
(Mutchler, 1967; Mutchler and Young, 1975). These authors found that
raindrop impacts are more erosive when the water depth is about one
fifth of the drop diameter and that the impacts are practically non
erosive when the soil is covered by water at a depth of about three
drop diameters or more.
Palmer (1963, 1965) also studied the effects of the impact of
waterdrops with the shallow water layer. He studied the stress
strain relationship on a surface covered by different water layer
thicknesses that were impacted by different drop sizes. His maximum
7
8
reported strain values occurred when the water depth was about one
drop diameter. Also, when the water depth was 20 mm (0.787 in.), the
stressstrain relationship was found to be about the same as without
the water layer. For a deeper water layer, the waterdrop impact
effects became negligible. He also reported that v/hen the depth of
the water layer was three times the drop diameter or more, the soil
loss was very small.
If the water layer is moving, then there is an additional ac
tive force creating soil erosion. Such overland flows usually have
very small depths and their mean velocities are not high enough to
produce erosive bed shear stresses. But the turbulence due to the
raindrop impacts does make the increased detachment and transport of
particles possible. This is because the raindrop impacts increase
the energy and the momentum transfer in overland flows.
In any rainfall event, the overland flow will be present after
the topsoil is saturated and the rainfall intensity exceeds the
infiltration capacity of the soil. Usually the depth and the veloci
ty of the sheet flow increase as the water moves downslope because
more rainfall is accumulated as the contributing area increases.
This flow tends to move towards microchannels in which the accelera
ting water tends to increase the scouring action forming rills or
small channels that usually grow in dimensions in the downstream di
rection. The rills may carry the water into bigger channels called
gullies where the now sedimentladen flow continues its erosive ac
tion. Finally, the water will reach a continuously flowing stream
9
such as a river. Here the soil will be carried until its final depo
sition in a reservoir, lake, delta, or ultimately, the ocean.
Usually a particle which is detached at the highest point of a
drainage area does not reach the river in the same rainfall event
since the erosion process usually is relatively a slow moving process
with respect to time. This is because the particle depends on an
external force (i.e., raindrop impacts, flowing water or wind) to be
detached and move downslope. If there is not any force capable of
moving the particle, it will remain in the same location. Since the
wind is excluded in this study, only during each rainfall event will
the particle have a downslope displacement. So the total distance
traveled by the soil particle will depend on the number of rain
storms, the specific rainfall characteristics and the soil character
istics.
The presence of rills and gullies in any area will depend on
the soil surface's properties, the steepness of the slope, and the
presence of vegetation. Therefore, one may find areas with highly
erosive soils in which gullies and maybe rills are absent due to the
lack of slope. On the other hand, it is possible to find gullies and
rills lacking even on very steep slopes if the soil is highly resis
tant to erosion (Ellison, 1947).
The raindrop impact effects in rills and gullies are usually
considered negligible compared to the flow discharge effects. This
is because the water depth in rills and gullies can be enough to sup
press the detachment capacity of the raindrop impact. In addition,
10
the flow discharge has enough velocity in itself to produce erosive
bed shear stresses. However, the raindrop impacts and the corres
ponding splashes are very significant in any area before the rills
are generated and the areas between the rills. These two areas are
usually referred to as the interrill areas.
Now, it is necessary to know the extent of the rainfall effects
on the soil erosion process. Young and Wiersma (1973) studied the
relative importance of raindrop impact and flowing water to the ero
sion process. This was accomplished by determining the source and
mode of sediment transport on a laboratory plot under conditions of
normal rainfall energy and greatly reduced rainfall energy. They
found that decreasing the rainfall impact energy by 89% without
reducing rainfall intensity, the soil losses decreased by 90% or
more. It was thus demonstrated that the impact energy of raindrops
is the major agent in soil detachment. For all three soils studied
80% to 85% of the soil loss originating in the interrill area was
transported to a rill before leaving the plot. Thus, it was indica
ted that the transport of detached particles from the plot was accom
plished mainly by flow in the rills.
From that study, Mutchler and Young (1975) found that the soil
carried along by the splash energy was only 10% to 17% of the soil
loss from interrill areas to rills. The remainder of the loss to
rills was carried in the thin surface flow which without raindrop
impacts carried little, if any, soil. Therefore, the conclusion was
11
that the raindrop impact was the driving force in transporting soil
in thin surface flows (sheet flows) to rills.
There are many considerations about the soil erosion process by
rainfall which have not been indicated in this section. It is better
to review them individually in order to understand this process from
single contributions of the factors and then joining them into a gen
eral soil erosion process description.
2.2 Initial Studies
Soil erosion has been studied extensively during the last half
century mostly due to its importance to agriculture and food produc
tion. Before the 1930s, the soil erosion problem was recognized but
not considered as a major problem. Therefore, there was not much
written about it and most of the literature available came from Euro
pean studies which did not apply directly to many of the conditions
found in the United States.
During the 1930s, there was an increased need for studies re
lated to soil erosion. It was realized that some of the most produc
tive lands were removed from agricultural production because the
water from rainfalls and the wind was carrying away the fertile top
soils and nutrients which the plants needed. Since there was not
much knowledge about the erosion process, the initial studies were
basically concentrated on collecting data which could help to estab
lish the magnitude of the problem and in studies to find some alter
natives or conservation practices to control erosion. In addition,
12
there were few studies in which the mechanism of soil erosion and
their effects were considered too.
Laws (1940) presented one of the first studies in which the
relation of raindrop size to erosion and infiltration rates were
considered. He also mentioned previous studies done by European and
American scientists around the turn of the century, and referred to
studies carried out at that time by himself and other researchers.
Ellison (1944, 1945, 1947, 1950) contributed a series of papers
in which he described the soil erosion process. It was the first
time this process was described and studied in such detail. Ellison
(1944) initially presented the current knowledge about the soil ero
sion process and the factors which might affect the process. He
developed an empirical equation for raindrop erosion (splash erosion)
based on the rainfall intensity, the diameter of the waterdrop and
the velocity of the drop. In 1945, he presented his experimental
results of the effects of raindrop impact and flow in the infiltra
tion capacity and the soil erosion. He divided the stuc(y in raindrop
effects alone, runoff effects alone, and the combined effects. Like
previous studies, many of his experiments were exploratory in nature
and the data had only qualitative significance.
Then Ellison (1947) proceeded to describe his approach to the
soil erosion problem step by step. He postulated that the soil ero
sion process was "a process of detachment and transportation of soil
materials by soil agents." This definition described the process as
13
composed of two principal and sequential events. In the first one,
the soil particles are torn loose, detached from the ground surface
and made available for transport, which is the second event. There
fore, the erosive capacity of any agent was comprised of two indepen
dent variables of detaching capacity and transporting capacity. The
raindrop impacts and the surface flow runoff were the erosive agents
he considered in his study. Wind was also recognized as an individu
al erosive agent, but not included in Ellison's research.
Ellison's approach was based on four different conditions
(i.e., detachment and transportation of particles due to raindrop im
pacts or surface runoff) to describe the soil erosion process. The
detachment of soil particles by the erosive agents was related to the
soil properties and conservation practices available to the area
under stuc(y. Meanwhile, the transport of soil particles by the ero
sive agent was considered to be a function of the transportability of
the soil, the intensity of the transporting agent, and the quantity
of soil already detached.
The effect of slope and wind were mentioned as sources of
splash transportation in Ellison's studies. The kinetic energy of
the runoff, the slope, the surface roughness, the thickness of the
water layer, and the turbulence generated by the raindrop impacts
were mentioned as parameters for surface flow transportation. How
ever, Ellison did not develop expressions to define each of these
parameters. More work and knowledge were necessary before the fun
damental relationships could be obtained.
14
Musgrave (1947, 1954) presented a review of the knowledge on
sheet erosion and the estimation of land erosion. Using data from
the available literature and from his experiments, he indicated that
the erosion was related to many variables expressed in the following
proportionalities.
Erosion ^ I30^
Erosion ^ S0*33
Erosion XL0,35
Area
where
I30 = maximum amount of rain in 30 minutes of rainfall
(inches)
S0 = slope gradient (percent)
X_ = slope length (feet)
He also presented the relative amount of erosion for different
vegetal covers. Adjustments between studied soils being exposed to
different rainfall, slope and slope length conditions were made in
order to present results of rate of erosion under a common basis. An
example of this procedure was presented in his 1947 study.
Ekern (1953) presented a good summary of the previous knowledge
and information needed about the rainfall properties that affect
raindrop erosion. Then he presented his approach to raindrop erosion
based on the kinetic energy of the natural rainfall and discussed the
15
rainfall parameters and soil factors needed to represent the erosion
process. He recognized the use of simulated rainfall as a tool for
obtaining a better understanding of the erosion process. However, he
emphasized the need for the control of the rainfall parameters (i.e.,
rainfall intensity, and drop size, pattern, shape and velocity) in
order to have the best representation of a natural rainfall while the
soil erosion data is collected.
Like Ekern, other authors have also discussed the use of simu
lated rainfall for soil erosion research. Among them, Meyer (1965)
and Bubenzer (1979) have presented detailed information about simula
ted rainfall conditions. The general consensus of all these studies
is that the drop size distribution, the drop velocity at impact and
the rainfall intensity are the basic parameters which need to be con
trolled and duplicated to the best possible accuracy.
In the next sections of this chapter, a review of the erosive
agents presented by Ellison (i.e., raindrop and surface runoff) are
presented in more detail.
2.3 Raindrop and Rainfall Characteristics
2.3.1 Raindrop Characteristics
It was mentioned before that the raindrop impacts are the ini
tial cause for detachment of soil particles from the bed surface;
they also provide the necessary turbulence to keep the particles in
motion in the shallow overland flows. Not all raindrops which impact
the soil surface during certain periods of time are identical. So,
it is necessary to study the raindrop characteristics in order to
16
understand the erosion process due to rainfall. Raindrop character
istics important in soil erosion are the drop mass, size, shape, and
their terminal velocity. Falling raindrops in air are not completely
spherical, but researchers have referred to an equivalent spherical
diameter De based on the actual mass of the raindrop to discuss the
variation in size between waterdrops.
Laws (1941) presented velocity measurements of waterdrops with
sizes ranging from 1 mm (0.039 in.) to 6 mm (0.236 in.) in diameter
falling through still air from heights of 0.5 m (1.64 ft) to 20 m
(65.6 ft). He also reported a few measurements of raindrop veloci
ties in order to compare with earlier observations. Lav/s' measuring
techniques consisted of a high speed photographic system, used to
measure the drop velocity and the flour pellet method to determine
the drop size. Laws' results showed that the waterdrops attained a
terminal velocity after falling a certain height. The height re
quired to reach terminal velocity increased as the drop size in
creased for drop sizes of about 4 mm (0.157 in.) or less. Beyond
that drop size the required height gradually decreased as the drop
size increased. The variations in the drop shape and the consequent
change in the friction resistance through the drop falling stage were
related to that reduction of the required height to reach terminal
velocity. Nevertheless, the terminal velocity always increased as
the drop size (i.e., drop mass) increased.
Later, Gunn and Kinzer (1949) presented what appears to be the
most accurate fall velocity measurements available. Using electronic
17
techniques to measure the fall velocity they were able to work from
drop sizes so small (about 0.75 mm = 0.029 in.) that the Stokes Law
was obeyed to up to (and including) drops large enough to be mechan
ically unstable (about 6.1 mm = 0.24 in.). This work was done under
controlled conditions in stagnant air at 760 mm Hg pressure, a tem
perature of 20C (68F) and 50% relative humidity. The new observa
tions resulted in generally larger values than those found by other
researchers but approached more to the values obtained by Laws
(1941). The new values were measurably smaller than Laws' values.
The overall accuracy of the drop massterminal velocity measurements
of Gunn and Kinzer's study was better than 0.7%.
There are other studies dealing with the behavior of the fall
ing raindrop. For instance, Blanchard (1950) studied the growth of
larger waterdrops caused by collision with small drops, the breakdown
of larger waterdrops and the deformation of the waterdrop with time.
Jones (1959) considered the shape of the raindrops during rainstorm
events and concluded that there was basically a mean shape which
varied consistently with the mass of the raindrop. However, he also
observed that the shape was the result of oscillations about a mean
and that the tilt observed in the raindrop's major axis was associa
ted with the wind speed and its prevailing direction in the atmos
phere at the moment the measurements were taken. Likewise, Jameson
and Beard (1982) studied the oscillating forms of the freely falling
raindrops. Epema and Riezebos' (1984) study indicated that the
18
oscillations are gradually damped and at terminal velocity their drop
shape observations showed that the drops attain equilibrium and have
an oblate shape. Comparison of the drop shape showed that the drops
obtained in the laboratory (still air conditions) were more oblate
than the drops of equivalent drop diameters in natural rain observed
by Jones (1959).
Some researchers have developed analytical approaches to des
cribe the raindrop size, shape, and falling speed. Spilhaus (1948)
assumed that a falling raindrop has an ellipsoidal shape. The sur
face tension effect was combined with the aerodynamic deformation of
the drop in order to maintain the steady shape and falling velocity.
His theoretical values partially agreed with Laws (1941) data, but
his approach was not able to describe the complex behavior of the
falling raindrop in air. McDonald (1954) presented a better analy
tical approach in which he concluded that under most conditions the
surface tension, the hydrostatic pressure and the external aerody
namic pressure were the three factors which had important roles in
producing the characteristic deformation of large raindrops.
Wenzel and Wang (1970) used a balance of forces approach to
stucty freely falling drops. That is, neglecting minor forces, they
considered the balance between the drag force, the buoyant force and
the gravitational force. Solving for the drag coefficient, CD, and
using data from Laws (1941) and Gunn and Kinzer (1949), they produced
diagrams for the drag coefficient of falling waterdrops in air. A
19
relationship between fall velocity and fall height was also obtained
using the balance of forces equation in an integral form.
Beard (1976) studied the waterdrop behavior in the atmosphere
dividing the analysis in three physically distinct flow regimes
1 pm < De < 20 pm with 106 < ReA < 0.01
20 pm < De < 1 mm with 0.01 < ReA < 300
1 mm < De < 7 mm with 300 < ReA < 4000
where
De
%
= equivalent spherical drop diameter
Vn Dp
= ~ = drop Reynolds number
Vq = drop velocity
VA = kinematic viscosity of air
For each regime he developed equations, using the drop size and the
physical properties of the drop and atmosphere, in order to estimate
the drop axis ratio, the projected horizontal drop diameter and the
terminal velocity.
2.3.2 Rainfall Characteristics
To evaluate soil erosion by rain it is necessary to know about
the rainfall intensity, the duration of the event, the size
distribution of the raindrops at a given intensity, and the kinetic
energy or momentum of the raindrops at a given intensity.
Laws and Parsons (1943) presented the drop size distribution
against rainfall intensity relationship. They used the mean raindrop
size, D50, as the value to represent the particle distribution
20
for a given rainfall intensity. The mean drop diameter was defined
as the abcissa of the point in the cumulativevolume curve having an
ordinate of 50%. Their empirical equation was presented as
D5q = 2.23I0,182 (2.1)
where D^q is in millimeters and the rainfall intensity, I, in
inches per hour.
They recognized that the raindrop size distribution at any
rainfall rate they presented was only an approximation. A variabil
ity of the drop size distribution from time to time for the same
rainfall intensity was also recognized and the possibility that a
similar raindrop size distributionrainfall intensity relationship
could be found elsewhere was mentioned too.
Chapman (1948) studied the effect of forest on the raindrop
size distribution and on the striking force at the soil surface. He
found that the volume of water striking the soil per unit area per
unit time in a pine plantation and in an open area were approximately
equal. The raindrop size distribution in the forest field showed a
more flattened shape instead of the bell shaped frequency reported by
Laws and Parson (1941) for open areas. This indicates that the
forested area had a more uniform distribution of the water volume
throughout the range of drop sizes. In addition, he observed that
the meandrop size in the open field increased with increasing rate
of rainfall, but for the pine area the mean drop was apparently
unrelated to the rainfall intensity (at least within the range of
rainfall rates measured). He also indicated that the raindrops
21
could reach again near terminal velocities in the forested area be
cause the soil did not have any other vegetation than trees. The
trees provided with 8.5 m (27.9 ft) of free fall distance between the
base of the canopy and the soil.
Other researchers (e.g., Mihara, 1951; Hudson, 1963; Carter et
al., 1974; McGregor and Mutchler, 1977; and Park et al., 1983) have
presented raindrop sizerainfall intensity relationships different
from the one proposed by Laws and Parson. Their basic differences
are considered to be due to the geographic location, climatologic
conditions, kind of rainstorm measured, the time at which the samples
were taken during the rainstorm events, and the method used to
measure the raindrop size.
The difference in the proposed empirical relationship can be
seen by presenting some of the proposed equations:
Carter et al. (1974)
D50 = 1.63 + 1.331 0.33I2 + 0.02I3 (2.2)
McGregor and Mutchler (1976)
D5q = 2.76 + 11.40 exp(1.40I) 13.16 exp(1.17I) (2.3)
Park et al. (1983)
D50 = 0.33 I0*12 (2.4)
where Djq has units of millimeters and I has units of inches per
hour except Park et al. who used I in millimeters per hour.
It should also be mentioned that Horton (1948) proposed a sta
tistical distribution of drop sizes at different spatial locations of
22
a storm in order to describe part of a thunderstorm model. Some fre
quency distribution curves to corroborate the applicability of the
model were also suggested.
However, due to the complexity of the rainfall process many
researchers have not followed Horton's approach. Instead they have
looked for the important parameters which might affect the soil ero
sion process and have concentrated their efforts on them. In terms
of rainfall effects, researchers have studied the energy and momentum
rainfall can provide to erode the soil surface.
The kinetic energy of the rainfall can be obtained from the
raindrop size distribution for the given rainfall intensity and the
terminal velocity for each raindrop size. Based on that approach
empirical equations have been proposed. Mihara (1951) proposed the
relationship
KEt = 21,400 I1*22 (2.5)
where KEt is presented as kinetic energy per unit area and time
[erg/(cm2 min)] and I has units of mm/10 min.
Wischmeier and Smith (1958) proposed the relationship
KEa = 916 + 331 Aog(I30) (2.6)
where KE^ is the kinetic energy per unit area [(ft ton)/(acre *
in.)] and I3q is the rainfall intensity corresponding to the
maximum measured 30minute rainfall intensity during the rainstorm
event with units of in./hr. This equation is used in the Universal
23
Soil Loss Equation which has been widely used to predict soil loss
from laboratory and field areas during the last thirty years.
Elwell and Stocking (1973) used the expression originally
developed in Hudson's masters thesis (1965) for their Rhodesia,
Africa, region
KEA = 29.82 12^51
where KEA has units of (J/m2) and I has units of mm/hr.
Carter et al. (1974) proposed the expression
(2.7)
KEa = 429.2 + 534.0 I3Q 122.5 I302 + 7.8 I303 (2.8)
where KEA has units of (ft tons)/(acre in.) while I30 follows
Wischmeier and Smith's (1958) definition and has units of in./hr.
McGregor and Mutchler (1977) presented their expression as
KEa = 1035 + 822 exp(1.22 I30) 1564 exp(1.83 I30) (2.9)
where KEA and I3q have the same units as Wischmeier and Smith (1958).
Kneale (1982) obtained for small rainfall intensities
(0.1 mm/hr < I < 7 mm/hr) the expression
log KEt = 0.90 + 1.25 I (2.10)
where KEt has units of J/(m2 hr).
Park et al. (1983) have proposed the expression
KEt = 211070 Cte I1,16
(2.11)
24
where KEt has units of J/(ha hr), I is in mn/hr and Cte is a tem
perature correction factor.
Rogers et al. (1967) discussed some of the sources of error in
calculating the kinetic energy of rainfall. They indicated that the
sources of errors are variations in the raindrop size distribution
even at different periods of the rainstorm with the same rainfall
intensity and the measuring technique used to measure rainfall inten
sity and wind effects.
Recently, Mualem and Assouline (1986) proposed an analytical
function to represent the raindrop size distribution which was cali
brated for Rhodesia (Hudson, 1965) and Washington, D. C. (Laws and
Parson, 1943) data. From it, the rainfall kinetic energy per unit
mass and the rainfall kinetic energy per unit time expressions were
presented as a function of rainfall intensity. The curves for rain
fall kinetic energy per unit mass differed significantly from known
empirical expressions obtained by other authors which used the same
data. Their rainfall kinetic energy per unit time curve was found to
have an insignificant deviation between both data places at low rain
fall intensities, but became noticeable at higher values of the rain
fall intensity.
Similarly, there are some relationships giving the momentum of
rainfall applied to a given surface and the rainfall intensity men
tioned in the literature. Elwell and Stocking (1973) used the
expression originally developed in Hudson's masters thesis (1965)
25
Ma = 75.3 1?5*2
(2.12)
where MA is rainfall momentum per unit area and has units of
(kg m)/(s m^) and I has units of nm/hr.
Park et al. (1983) proposed the use of the expression
Mt = 64230 I1'09
(2.13)
where Mt is the rainfall momentum per unit area per unit time
(kg m/s)/(ha hr), I has units of mm/hr and Ctm is the tem
perature correction factor.
Finally, the relationship between the total number of drops
collected in a unit area per unit time, Ndr0p [drops/im2 s)],
and the rainfall intensity I (mm/hr),
drop 154 I05
(2.14)
presented by Park et al. (1983) may also help in the future to im
prove relationships for the soil erosion process.
All of these equations presented here have certain conditions
in order to be used correctly. The reader is referred to the origin
al studies for more information.
2.4 Splash Erosion
2.4.1 Waterdrop Splash
The study of the waterdrop impact and the consequent waterdrop
splash was improved with the introduction of the high speed cameras.
With this equipment, the different conditions and the time sequence
of this process were studied in detail. Ellison (1950) originally
presented sequences of photographs about the waterdrop splash on soil
26
surfaces with different water layer thicknesses over the bed surface.
The variation in splash characteristics with respect to changes in
the water layer thickness was visually explained in those photo
graphs.
But it was not until the late 1960s that the interest on the
waterdrop splash process and splash sequences were really studied
thoroughly. Mutchler authored and coauthored a series of articles
in which the individual characteristics of the waterdrop splash were
presented.
Mutchler (1967) studied the waterdrop splash at terminal velo
city over different types of surfaces with and without a water layer
covering it. He studied the effects of the drop diameter, the water
depth, the roughness and the softness or hardness of the solid sur
faces on the splash characteristics. A set of parameters were estab
lished to describe the geometry of splash. For this he used the
width of the crater of the splash, the height of the splash sheet
wall, the radius of curvature of the splash sheet wall, the angle at
which the sheet wall goes with respect to the water surface, and the
angle at which the splash droplets are ejected from the splash sheet
wall. Since these parameters changed their values with respect to
time he used the characteristic shape occurring at the time of maxi
mum sheet wall height to show the effect of the water layer depth on
the splash. He concluded that the water depth had its greatest ef
fect on the waterdrop splash at depths of about onethird of a drop
diameter and that the splash geometry changed very little at water
depth greater than one drop diameter.
27
Mutchler and Hansen (1970) used data from Mutchler (1967) to
develop empirical dimensionless equations to represent the waterdrop
splash. They used the water layer depth to drop diameter ratio
(h/De) as the only parameter needed to obtain the other dependent
parameters already presented by Mutchler (1967).
Mutchler (1971) also presented relationships for the splash
droplet production by waterdrop impacts on a glass surface with a
water layer over it. Using the h/De ratio, he presented relation
ships for the number of droplets produced by one waterdrop impact,
the mean droplet diameter size of the droplet size distribution and
the standard deviation of that distribution. Discussions of how
these parameters changed with the water layer depth and the waterdrop
size or weight were also present.
Mutchler and Larson (1971) studied the amount of splash that a
waterdrop at terminal velocity could produce by impacting a water
layer over smooth glass at various water depths. They presented
empirical equations to predict the weight of water splashed which
indicated a maximum splash weight at h/De of 0.14 and 0.20 for De
equal to 5.6 irni and about 3 mm, respectively. The influence of
greater depth became relatively insignificant at a depth of about
threedrop diameters. They stated that without splash there cannot
be splash transport. At shallow water depths splash weights greater
than two times the waterdrop weight were observed. But as the water
depth increased to three waterdrop diameters or more, the splash
amount decreased and became relatively insignificant.
28
Mutchler and Young (1975) presented a relationship for the rate
of change of width of the crater with time. From this, they obtained
an expression of the lateral (horizontal) velocity, u^ of the wa
ter moving away from the impact site along the surface. They also
obtained a rough estimate of the viscous bed shear stress t0 by us
ing the equation
where
du = rate of change of the horizontal velocity
dz = increment of vertical distance in the water
p = dynamic viscosity of water
Based on these conditions an estimate of the minimum velocity
required to detach soil particles from the surface and how long those
shear stresses could last before they become smaller than the criti
cal shear stress, tcr was presented.
From this approach Mutchler and Young were able to show that
the erosive action of a waterdrop impact was effective very early
after impact and thus in the vicinity of the center of impact. They
also showed that for water layer depths equivalent to threedrop di
ameters, the soil is essentially protected from raindrop impacts.
Finally, it was also indicated that most of the water splashed from
the area of impact came from the water layer and not from the water
drop itself.
29
Contemporary to Mutchler's works, Hobbs presented another ser
ies of articles about waterdrop splash characteristics. Hobbs and
Kezweeny (1967) measured the number of droplets produced by the im
pact of a waterdrop on a water surface and the electric charge of
these droplets. The number of droplets produced was a function of
the fall distance of the waterdrop. A fall distance of 10 cm (3.94
in.) or less did not produce any splash, and for fall distances up
to 200 cm (78.7 in.) the number of droplets was found to increase
linearly with the fall distance. Mutchler's results cannot be com
pared with the results obtained by Hobbs and Kezweeny because the
latter study did not test fall distances higher than 200 cm (78.7
in.). Consequently, no terminal velocity of the waterdrop was
reached in this latter study. It should also be mentioned that the
latter study reported that nearly all of the spray droplets carried a
negative charge and for the range of fall distances used it appeared
that the fall distance had little effect on the charges carried by
the spray droplets.
Hobbs and Osheroff (1967) and Macklin and Hobbs (1969) also
studied the effect of the water layer depth on the waterdrop splash
but their major interest was the study of the Rayleigh jet produced
by the returning (converging) fluid filling the crater created by the
waterdrop impact.
The waterdrop splash has also been studied analytically using
the NavierStokes equations. Each study has established its assump
tions and boundaries to the problem resulting in simplified equations
which are solved by numerical analysis and computers.
30
Harlow and Shannon (1967a, 1967b) solved the NavierStokes
equations for the waterdrop impacting a water layer phenomenon by
neglecting the viscosity. Also, surface tension effects at the water
surface were not considered. Solutions were presented for waterdrop
impacts onto a flat plate, into a shallow pool and into a deep pool.
The changes in splash configuration with respect to time were pre
sented for each case. Information about the pressures, velocities,
droplet rupture and effects on compressibility were also presented
for each water depth studied.
Wenzel and Wang (1970) used a different numerical approach than
the one used by Harlow and Shannon (1967a, 1967b) and included the
surface tension. Their results only consider the initial stages of
the waterdrop impact into stagnant water due to limitations in the
time of execution of the program and economic restrictions. These
initial stages included the period of time in which the waterdrop
impacts the water layer and the water moves radially outward. The
inward direction motion of the water was not included in the study.
Their results included a maximum impact pressure model and a quanti
tative discussion of pressure distribution, boundary shear, surface
tension effect, free surface configuration and various forms of ener
gy and their transformation during the impact process. Theoretical
results from the computer solution were successfully verified with
their experimental data of the impact pressure at the bottom surface
of a pan at various water layer depths. They used waterdrops of
various sizes falling at different impact velocities. Wenzel and
31
Wang also showed that surface tension cannot be neglected in this
kind of study.
Huang et al. (1982) neglected the body (gravity) force, the
viscosity forces and the surface tension in order to examine the
raindrop impact on a smooth rigid surface. The maximum pressure was
reported to occur at the contact circumference and that the lateral
jet velocity at the rigid surface was considered to provide the cru
cial mechanism in the raindrop soil detachment process. From this
they implied that the three critical factors important in defining
the soil resistance against the raindrop impact were the soil defor
mation characteristics, the soil shearing strength, and the soil
surface microrelief.
Then, Huang et al. (1983) presented the deformation pattern of
a solid material under a raindrop impact by numerical analysis of the
assumed linear elasticity material. The deformation due to imposed
impact loadings of: (a) a steac(y uniform load, and (b) a simulated
raindrop impact load were compared and found to be completely differ
ent. As an example, for the material with a low modulus of elasti
city, a uniform depression was found under steady, uniform load,
while a coneshaped depression was shown under the simulated raindrop
impact. As the authors indicated, this study only presented the
shape of surface deformation, but the interaction between the lateral
jet stream and the irregularities of the soil surface were believed
to be the ones which determined the amount of splashed soil.
32
Recently, Wright (1986) presented a physicallybased model of
the dispersion of splash droplets from a waterdrop impact on a slop
ing surface. He considered the forces and momentum transfer acting
at the moment of impact in order to obtain the velocity vectors of
the droplets. The absorption of some of the waterdrop's momentum by
the soil particles was considered as well as the air resistance ex
erted on the droplets while they travel in the air. The effects of
slope, wind, raindrop size and some soil properties on the droplet
distribution were also included. The probability of a particular
droplet size being transported was obtained from splash droplet size
distribution obtained from Mutchler studies. Although the proposed
model considered the soil absorption of the waterdrop momentum the
model does not consider the detachment of soil particles which would
be the next stage toward a model of soil erosion by rainsplash.
2.4.2 Splash Erosion Studies
The literature shows many studies dedicated to the splash ero
sion. There are studies about: techniques used to measure the
splash erosion, soil and rainfall properties which are important in
this process, mechanics of the process, rate of soil detachment with
respect to time or to rainfall intensity, empirical relationships to
represent the erosion rate of this process, etc. Not all of the pub
lished studies can be presented here but at least a brief description
of the current stage of this erosion process is presented.
The most popular method used to measure the splash erosion con
sists of exposing a small amount of soil in a cup to the direct
33
impact of falling waterdrops with a known rainfall intensity. The
amount of soil material that has been removed from the cup after a
certain period of time is considered to be the soil loss due to
splash on that soil material. Ellison (1944) and other studies have
considered that the soil detachment at a given rainfall intensity
decreases as time increases, while Bisal (1950) and others indicate
that it is constant. The latter authors consider that the decrease
in detachment rate is due to the obstruction created by the cup's
wall as the soil surface progressively decreases with time and have
proposed correction factors for the use of the splash cup technique
(Bisal, 1950; Kinnell, 1974). Farrell et al. (1974) has also pro
posed a correction factor for the geometric parameters (i.e., size
and shape) of the soil containers used to measure splash erosion.
The splash erosion has been related to soil characteristics
(i.e., particle size distribution, presence of aggregates, organic
content, and others), the bed slope, and the rainfall characteristics
(i.e., rainfall kinetic energy (or rainfall momentum) per unit area
and time, drop size, shape, and impact velocity). From it, each
study has presented empirical equations to predict the amount of soil
splash, detached or transported from a given surface area (Ellison,
1944; Ekern and Muckenhirn, 1947; Ekern, 1950; Bisal, 1960; Bubenzer
and Jones, 1971; Quansah, 1981, and Gilley and Finkner, 1985).
The use of the rainfall kinetic energy or the rainfall momentum
in those empirical equations appears to be a preference of the au
thors. However, some of these studies have presented their
34
experimental data or statistical foundations to support the use of
their rainfall parameter in their equation. Rose (1960) justified
the use of rainfall momentum per unit area and time instead of using
the kinetic energy per unit area and time. Meanwhile, Gilley and
Finkner (1985) presented statistical analysis which indicates that
the kinetic energy times the drop circumference is better. Apparent
ly the literature shows that there is a majority of studies prefer
ring the rainfall's kinetic energy more than the rainfall's momentum
for the development of their splash erosion equations, but the use of
any of these two rainfall parameter must be physically justified in
each case.
Bubenzer and Jones (1971) also studied the effects of drop size
and impact velocity on the splash detachment. They found that small
er drops produced less splash than the larger ones even though the
kinetic energy, the total rainfall mass and impact velocity were
almost constant. Therefore, more parameters are needed to describe
the splash erosion.
The effect of the bed slope is also very important in the
splash erosion (e.g., Ekern and Muckenhirn, 1947; Ekern, 1950; Free,
1952; DePloey and Savat, 1968; Savat, 1981, and others) because the
soil downslope transport increases as the bed slope increases. Free
(1952) also indicated that the effect of the slope in relation to the
direction of the storm was important in determining the amount of
soil removed from the soil pans. Losses from pans facing the direc
tion of the storm were found to be three times those from pans facing
35
away from the direction of the storm. They indicated that this is
due to the fact that the normal component of the raindrop increases
if the bed slope is facing the direction of the storm.
Mazurak and Mosher (1968, 1970) and Farmer (1973) have reported
that for any soil grainsize there is a linear relationship between
the soil detached by raindrop impacts and the rainfall intensity.
Mazurak and Mosher studies were conducted by separating the soil par
ticles or aggregates in ranges of sizes and testing each of them in
dividually, while Farmer's stucty was for the mixture of sizes. In
these studies the curves of soil detachability against particle size
had a bellshaped form with a peak around the 200 pm size. Ekern
(1950) also found that fine sand (175 pm to 250 pm) gave the largest
amount of soil transported. Farmer's results showed curves skewed
toward the smaller sizes with the tendency to be a more skewed curve
as the content of smaller sizes increased in the original soil. In
addition, Fanner's stucty included some overland flow effects which
changed the susceptibility to detach soil particles by raindrop im
pact. Without overland flow the soil particle sizes in the range of
110 pm to 1450 pm were most susceptible to detachment by raindrop im
pact, with the peak range from 238 pm to 1041 pm. Meanwhile, with
overland flow the most susceptible size range was 219 pm to 2034 pm,
and the peak ranged from 440 pm to 1336 pm.
A previous study by Rose (1960, 1961) showed that soil detach
ment by raindrop impacts and the rainfall intensity was not linear.
This departure of linear characteristic was associated with the
36
resulting breakdown of the structure of the aggregates in the soil by
the raindrop impacts.
DePloey and Savat (1968) used autoradiographies of radioactive
sand to study the splash mechanism. Their results showed the impor
tance in the splash phenomenon of the grainsize distribution of
sands, the slope gradient, the angle of ejection, the distribution of
grains around the point of impact of the raindrops, the characteris
tics of the rain, and the physical properties of the soil. Using
their data and physical considerations in developing a mass balance
of the soil particles, they were able to describe the splash mechan
ism for horizontal surfaces, sloped surfaces, and for segments of a
convex slope.
Morgan (1978) indicated that his results of rainsplash erosion
from field studies of sandy soils confirmed the relationships between
splash erosion, rainfall energy, and bed slope obtained in laboratory
experiments by other researchers. He also reported that only 0.06%
of the rainfall energy contributed to splash erosion and that the
major role of the splash process is the detachment of soil particles
prior to their removal by overland flow.
Poesen (1981) studied the erodibility of loose sediments as a
timedependent phenomenon. He indicated that the variations in the
detachability of soil particles during the rainfall event could be
explained by changes in water content (including the liquifaction and
the development of a water layer on the surface), cohesion and granu
lometric composition of the top layer. In his case the presence of
37
a water film (less than one raindrop diameter size thick) decreased
the detachment. So he concluded that findings by other researchers
about the increase in detachment when the thin water layer was pre
sent was not a universal phenomenon but that it might be limited to
the materials and procedures used in each experiment. The relation
ship of amount of soil detached by splash to the mean grain size had
very similar shape to the relationships obtained in previous studies
by Ekern (1950) and Mazurak and Mosher (1968). This relationship was
reported to be very similar to the relation between grain size and
the susceptibility to runoff and wind erosion established by other
researchers. For this study the highest detachability was found to
be for the fine very wellsorted sand with a mean grain size of
96 um.
Yariv (1976) had also considered the presence of water as a
very important factor when he presented the concepts for a theoreti
cal model to describe the mechanism of detachment of soil particles
by rainfall in three stages: dry soil conditions, soilwater mixture
conditions and soil with overland flow conditions. The model was
proposed as a single general equation with changing coefficient val
ues in order to describe the three stages of the mechanism. A sto
chastic point of view was suggested by Yariv for the solution of the
model.
Savat (1981) presented results of splash erosion in which net
discharge of sediment (downstream splashupstream splash) was found
to increase proportionally to (sin9 )0,9 with respect to the
bed's inclination with horizontal. This sediment discharge was also
38
associated with the susceptibility to splash erosion of the soil
which, like other previous researchers have indicated, is a complex
function of the moisture content of the sand and its grainsize dis
tribution. A technique was proposed in order to obtain the mean pro
jected splash distance along the sloped plane surface.
Park et al. (1982) used dimensional analysis to model the
splash erosion of the two possible domains; the direct impact or drop
solid domain, and the dropliquidsolid domain. The dropsolid do
main was described with analytical relationships from the conserva
tion of momentum on a sloped bed. For the dropliquidsolid domain
the water layer depth effects were related to the erosion rates by an
exponential form. The results showed that the dropsolid domain is
time dependent while the drop1iquidsol id domain is independent of
time. Bed slope effects were also considered in both domains.
Recently, Riezebos and Epema (1985) presented the importance of
drop shape on the splash erosion. They found that for all test com
binations together, the introduction of the observed drop shape in
erosivity parameters only produced minor improvements in the relation
between erosivity and detachment (or transport) by splash. However,
when they used small fall heights and low fall velocities, as in many
rainfall simulators and drop tests, the prolate drops produced a
splash detachment which was two to three times higher than the one
produced by oblate drops at impact. This was partly associated with
the high splash erosion in areas below the vegetation.
39
The presence of vegetative cover or any manmade cover over the
soil will reduce the splash erosion because this cover will absorb
most of the raindrop energy (Mihara, 1951; Free, 1952; Young and
Wiersma, 1973, and others). The cover prevents surface sealing; con
sequently, the infiltration is not drastically reduced and the rate
of runoff is diminished. Free (1952) reported that the presence of
straw mulch reduced splash loss to about onefiftieth (1/50) of that
from bare soil and sheet flow losses to onethird (1/3).
Osborn (1954) indicated that, in addition to the already men
tioned soil and rainfall characteristics, the land use management and
conservation practices also affect the splash erosion. Other soil
characteristics not mentioned before were also related to this ero
sion process.
2.5 Overland Flow Erosion
2.5.1 Hydraulics of Overland Flow
Knowledge of the hydraulics of the surface water runoff is
needed for the detailed understanding of the general soil erosion
process. Many studies have been completely dedicated to this complex
overland flow hydraulics. Surface runoff is the most dynamic part of
the response of a watershed to rainfall.
The runoff from a watershed can be subdivided in sheet flow;
rills and gullies flow; and openchannel flow. Overland flow deals
basically with the first two kinds of flows and it is the one which
supplies water and sediment to the open channels. The equations
40
used to describe the open channel hydraulics can be used to describe
the overland flow hydraulics if additional terms are included in or
der to account for the rainfall effects, the slope effects, and the
very shallow flow conditions of overland flow.
The most frequently used hydraulic parameters associated with
overland flow with rainfall are water depth (h), mean flow velocity
(0m), or discharge flow rate per unit width (q). These parameters
have been related to the detachment and transport of soil particles
in many empirical or semiempirical approaches. Usually, the studies
are based on the correlation between the total flow discharge at the
lower end of the area under study and the total soil loss of the
area. Other studies have also considered the rate of change of water
and soil loss with respect to time. Additional parameters which have
been used in overland flow erosion studies are bottom shear stress,
pressure at the bottom of the shallow water flow, roughness of the
bed surface (with and without vegetation), longitudinal slope of the
bed surface and its longitudinal length.
Rainfall provides input of water over the area. Depending on
infiltation, this may allow the flow discharge to change as the water
flows downslope. For this reason this flow is usually called spa
tially varied flow. The theory used to describe this flow is based
on the principles of mass and momentum conservation. Keulegan (1944)
first derived the onedimensional dynamic equation for two dimension
al spatially varied flow considering the rainfall as lateral flow.
41
Other derivations for this flow condition can also be found in Chow
(1959, 1969), Grace and Eagleson (1965, 1966), Chen and Hansen
(1966), Chen and Chow (1968), Yen and Wenzel (1970), Morgali (1970),
Kilinc and Richardson (1973), and others. The presentation of terms
in the equation may change depending on the assumptions and boundary
conditions used in each study.
In general form, the onedimensional conservation of mass equa
tion for spatially varied flow for sloped beds was presented by Chen
and Chow (1968) as
+ ^ = (I f) B cos 9 + q.
at 9x L
(2.16)
The corresponding dynamic equation was presented as
9(Aum) + 9(gAum ) _
at
ax
Bj I VD B sin( 0 + fi) BL 0mqL
(2.17)
= g A sine g A S, g i[A(h cos e + h.)]
3X *
where
A = crosssectional area
f = infiltration rate
t = time
x = longitudinal distance
B = width of the channel section
fi = angle of the path of falling raindrops with respect to the
vertical axis
= momentum correction factor for the flow velocity distribu
tion
B
42
BÂ¡ = momentum correction factor for the raindrop's velocity
distribution
B_ = momentum correction factor for the lateral flow velocity
distribution
qi_ = lateral flow discharge per longitudinal unit length
g = gravitational acceleration
h* = overpressure head induced by the raindrop impacts over the
hydrostatic pressure head.
Sf = friction slope
The overland flow equations which are a special case of the
channel flow, can be readily obtained by considering the discharge
per unit length, q = Am/B. Other considerations are that the
lateral flow vanishes, B = 1, and the area becomes A = h(1) = h.
Hence, q = 0mh and the equations are expressed as follows:
 Conservation of mass for overland flow
8h + 9mh) = d f) cos 6 (2.18)
at ax
 Momentum equation for overland flow
9(hm) +
at
a(Bh^m )
ax
gh(S Sf) g ^ [h(*. h cos e + hj]
(2.19)
+ 8i I Vd si n(0 + fl)
which for nearly horizontal beds gives S = sin e= tan e= SQ
43
Figure 2.1 shows a sketch defining these overland flow parame
ters. The right hand side terms of the continuity equation are the
sources of water. For the sheet flow case, the assumption of water
flowing in parallel streamlines toward the rills makes the lateral
flow term (q^) equal to zero, but for rills or any open channel flows
the lateral flow must be considered. The infiltration term is zero
if the bed surface is zero. For overland flow studies researchers
have also used the term rainfall excess, I* = rainfall intensity
minus infiltration rate. Overland flow will not exist if rainfall
excess shows a negative or zero value. This term was proposed by
Eisenlohr (1944) in a discussion of the one dimensional dynamic equa
tion derived by Keulegan (1944) for overland flow.
The terms in the momentum equation, or the socalled dynamic
equation (Equation 2.19) in order of sequence from left to right have
the following significance: (a) the unsteady term or local accel
eration term; (b) the convection acceleration term; (c) the force due
to the water weight and the friction loss term or boundary shear
force term; (d) the pressure gradient term which includes the rain
fall overpressure term; and (e) the momentum influx due to the fall
ing raindrop's component in the longitudinal slope direction. Some
studies have neglected the overpressure term but have included the
q. _
term 8(1 f + _L)Um to account for the retarding effect of rainfall
B m
excess and lateral inflow due to the mixing of the additional mass.
This term is obtained when the continuity equation is multiplied by
B0m and introduced in the momentum equation as a substitution of
the convection acceleration term.
44
Figure 2.1. Definition Sketch (based on Chen and Chow, 1968).
45
The literature presents studies in which the continuity equa
tion and momentum equation are used for overland flow descriptions
based on different assumptions and boundary conditions. There are
studies for cases of steady or unsteady state conditions; flows over
porous or impervious surfaces; with socalled physically smooth or
rough boundaries; under laminar or turbulent conditions; with fixed
or loose boundaries, and with or without wind effects. In most stud
ies the momentum correction factor 0 was assumed equal to unity due
to the difficulties in obtaining the velocity distribution of the
shallow overland flow. The use of 0 = 1 assumes uniform velocity
distribution in the cross section. Usually the momentum influx due
to rainfall (last term in Equation 2.19) has been neglected. This
term may be important in cases of steeper slopes or under windy con
ditions (Rogers et al., 1967, and Yoon, 1970). Consequently, the
qualitative judgment of the results of each study must be based on
the assumptions and methodology used by the authors. The possible
general application of the results should also be restricted by the
same considerations.
The study of the relative importance of each term in Equations
2.18 and 2.19 may help to simplify these equations and allow the de
velopment of simple hydraulic models based on these physical princi
ples. Table 2.1 presents the range of values of variables and dimen
sionless parameters in overland flow as reported by Grace and Eagle
son (1965). These values were obtained from an extensive literature
search in order to establish a similarity criterion for the modeling
of overland flow.
Table 2.1 RANGE OF VALUES OF VARIABLES AND SOME DIMENSIONLESS PARAMETERS IN OVERLAND FLOW
Model
Prototype
Variable
Units
Range
Typical
Range
Typical
Velocity, U
m
ft/sec
0.05 to 1
0.2
0.1 to 4
0.05
Depth, h
ft
0.005 to 0.05
0.02
0.02
Overland flow
ft
1 to 50
1.5
200 to 3000
1000
length, Le
Flow Reynolds number,

25 to 1000
100
100 to 20,000
800
Ref = Â¡Â¡mh/v*
 2 2*
1 +3 **
1
2 3
1
gh/um = 1/Fr

0(10 ) to 0(10 )
0(10 )
0(10 ) to 0(10 )
0(10 )
B


o
o
H
o

o
o
H
o
Friction coefficient, Cf
_
0(102) to 0(10_1)
__
0(10"3) to 0(102)
_
h/Le

0(104) to 0U0'1)
0(102)
0(106) to 0(10"3)
o
*
o
1
cn
I/m

o(io5) to odo"1)

0(106) to 0(10"3)

V^m

0(10) to 0(103)

0(10) to 0(103)


0(106) to 0(10~3)

Fr = Froude Number and v = kinematic viscosity of water.
** (10n) = Order of magnitude of 10n.
'ir'ir 4"
No infiltration in the model.
Source: Grace and Eagleson (1965, 1966).
47
Robertson et al. (1966) and Yoon (1970) also presented the
momentum equation for the case of steady spatially varied flow over
an impervious surface with mild slope and discussed the significance
of each term of their equation. Both studies used almost the same
assumptions and presented the momentum equation in the form
2
3h (1 Bq ) = S Sf ZBlq + 1 VD cos a (2.20a)
9* 9^ gh2 9h
I T 1 U I 1
Sj = S Sf S2 + S3 (2.20b)
in which B = 1 for Robertson et al. study, and Sj, S2 and S3 repre
sents the simplified form of each term in Equation 2.20a.
Table 2.2 shows the relative magnitude of the terms S, Sj, S2
and S3 with respect to Sf. These values indicate that the most
significant terms of the momentum equation are S = sin6 and Sf and
the remaining terms are at least two orders of magnitude smaller than
S or Sf. The contribution of these less significant terms (S^, S2
and S3) showed fluctuations which were due to the different testing
conditions at the time the measurements were collected (i.e., rain
fall intensity and bed slope).
There are studies of overland flow with rainfall (e.g., Grace
and Eagleson (1965, 1966) and Chen and Chow (1968)) which have indi
cated that the pressure distribution is not hydrostatic. They have
used an overpressure term in the momentum equation in order to
Table 2.2 RELATIVE MAGNITUDES OF THE TERMS S, S^ S2, AND S3 IN TERMS OF Sf
Reference
so
Relative Magnitude (percent)
S
sf
S1
sf
s2
Sf
S;
3
F
Min
Max
Min
Max
Min
Max
Min
Max
Yoon
0.005
100.2
122.2
0.467
12.68
0.445
15.72
0.0112
1.157
Yoon
0.01
99.0
110.8
3.45
3.58
0.328
13.29
0.0116
1.193
Robertson et al.*
0.05
100**
0.144
0.27
1.08**
0.612
1.82
Robertson et al. (1966) results are based only on two sets of absolute value reported in their article.
They used Sf = S = sinwhile Yoon (1970) obtained Sf using Equation 2.20b.
**0nly one value was reported.
4=.
00
49
account for the increase in pressure due to the raindrop impacts and
the vertical momentum influx of the raindrops. This overpressure
term was presented by Chen and Chow (1968) as
P* = BÂ¡ p I VD cos0 cos (0 + fl) = p g h^ (2.21)
where
P* = overpressure due to raindrop impact
p = fluid density
They considered that this overpressure was uniformly distribu
ted over the cross section except at the free surface where P* be
comes zero in order to have atmospheric pressure at the free surface.
This approach is based on the assumption that the total head, T^,
over a vertical cross section is constant or
2
Th = y cose + + a = constant (2.22)
Y 2g
where
P = y(h y) cose + h^
a = energy correction factor
y = distance from the bed surface to a location in the water
This assumption created a discontinuity in pressure at an infinites
imal small distance, Ay, from the free surface.
Grace and Eagleson (1965) have considered that the overpressure
distribution was linearly distributed from zero at the free surface
to a maximum value of 2P+ at the bed surface. They presented expres
sions for the overpressure term based on vertical momentum equation
50
and the order of magnitudes of each term. For the horizontal bed
with no infiltration and vertical falling raindrops the overpressure
term becomes
P+ = 0.5 p I VD (2.23)
Other researchers have used the overpressure term when the
momentum equation was presented in their studies (e.g., Kisisel,
1971; Kilinc and Richardson, 1973; and Shahabian, 1977). Kisisel and
Shahabian studies also included the rainfall turbulence effect in h*
and following Grace and Eagleson's approach of linear overpressure
distribution the h+ expression was presented as
h* = g [ \ BI 1 VD COS0 C0S (e + + v'2(h)^ (2.24)
where
2
v1 (h) = variance of vertical velocity fluctuations at the
free surface.
However, Shahabian's results show that the overpressure term induced
by the momentum influx of the raindrops seems to be a constant addi
tion to the hydrostatic pressure except at the free surface where
both the hydrostatic and overpressure terms are zero. This was based
on measurements at locations between 0.05 to 0.70 the water depth.
The magnitude of this overpressure with respect to other terms
in Equation 2.19 is sometimes small and the overpressure term is usu
ally neglected. The other reason to neglect this term is the collec
tion of data for the evaluation of h* in special values of Vq
and O.
51
It should also be mentioned that the calculated P* value is an
average pressure magnitude which is uniformly distributed over a
large surface area (with respect to the raindrop impact area) and
time, while the falling raindrops reached the overland flow randomly
in space and time. Therefore, the maximum overpressure due to the
rainfall will be larger than and very localized in space and
time.
Palmer (1963, 1965), and Wenzel and Wang's (1970) data present
some of the rainfall effects on the pressure at the bed surface which
is protected by a water layer. Unfortunately, the data were collect
ed from stagnant water and overland flow effects were absent. No
pressure data which might include rainfall effects and overland flow
effects were found in the literature review of the present work.
2.5.1.1 Simplified Solutions, the Kinematic Wave Method
Due to the complexity of the solution of the longitudinal mo
mentum equation (Equation 2.19) with all of its terms, the research
ers have used some assumptions and simplifications in order to obtain
the magnitudes of the hydraulic parameters needed to describe the
overland flow with rainfall. One of the simplest and most frequently
used approach is the kinematic wave method.
The kinematic wave method has been applied to overland flow
over a sloping plane in many studies with good success as an approxi
mation of these flow conditions (e.g., Lighthill and Whitman, 1955;
52
Henderson and Wooding, 1964; Wooding, 1965a, 1965b, 1966; Woolhiser,
1969; Eagleson, 1970; Morgali, 1970; Muzik, 1974; Li, 1979; Lane and
Shirley, 1982; Crowley II, 1982; Rose et al., 1983a, and others).
This approach uses the continuity equation for unsteady spatially
varied flow (Equation 2.18) and a simplified momentum equation in
which all terms, except bed slope (S = sine) and friction slope (Sf)
are neglected. This is based on the low numerical significance of
these terms in comparison to the magnitude of S and Sp From this
it is obtained that S = Sp A relationship between flow discharge
per unit width (q) and the flow depth is established by
(2.25)
where
a^ and b^ are coefficients expressed by the following
Laminar Flow bk = 3
bk = 5/3
(using Mann
ing's equa
tion in
English units)
Turbulent Flow
equation)
where
v = kinematic viscosity
Nm = Manning's roughness coefficient
53
C = (8g/Cf)0,5 = Chezy's coefficient
Cf = Darcy Weisbach's friction factor based on pipe diameter
Eagleson (1970) reported that experimental data from Horton
(1938) showed that the bk value was about 2.0 for natural surfaces,
and that further studies had supported that value for different kinds
of surfaces (e.g., vegetated surfaces, clipped grass, and tar and
gravel). The fluctuations of the bk exponent had been associated
with roughness effects. Usually an increase in roughness is associa
ted with the increase of the water depth which means a decrease of
the exponent's value. Muzik's (1974) results showed that bk was
exactly 1.66 = 5/3 for a galvanized sheet metal surface treated with
a diluted solution of hydrocloric acid to change the nonwetting
metal surface into a wetting surface.
The value of a^ is obtained based on known values of NM or C.
Woolhiser (1975), Lane and Shirley (1975), Podmore and Huggins
(1980), Engman (1986) and others have presented tables of typical
values for Manning's and Chezy's C coefficients which can be
used in overland flow studies.
The method of characteristics is frequently used to solve the
kinematic wave equations because it only has a single characteristic
relation to solve, namely,
k m
(2.26)
since
(2.27)
54
Using this method, Henderson and Wooding (1964) proposed a
series of relationships which allowed calculation of the surface run
off from a sloped bed surface at any location along the bed surface
and at any time. The method can also be used to produce the hydro
graph at any point along the sloped plane.
When the kinematic method is used for watershed modeling, the
watershed is divided in segments with constant slope and the water
flow is routed along the watershed segments (Woolhiser, 1975). Wool
hiser (1969) also used the kinematic approach to model the overland
flow on a converging surface on which the water moved toward a center
point in a radial motion.
Morgali (1970) presented computer solutions to this method and
studied the behavior of the equations for both cases laminar and tur
bulent flows. The variation of the flow regime along the sloped bed
was also considered if rainfall and bed surface conditions were
favorable and enough time for the test was allowed. His hydrograph
results agreed very well with the observations. The only discrepan
cies were observed on the rising segment of the hydrograph after the
inflection point of the rising limb and before the equilibrium flow
was reached at the downstream end of the bed surface. The reason for
this is that the kinematic approach does not predict that inflection
point in the rising limb.
Muzik (1974) tested the kinematic wave method against the in
stantaneous unit hydrograph method under laboratory controlled over
land flow due to rainfall conditions. He concluded that runoff from
55
an impervious surface generated by rainfall is a highly nonlinear
process and any linear analysis of the process does not strictly ap
ply. Linear models could only be used as a linear approximation of
the rainfallrunoff relationship. On the other side, the kinematic
wave model was able to better represent the rainfallrunoff rela
tionship and predicted values which agreed very well with the ob
served values. The model responded very well to changes in rainfall
intensity and slope of the runoff plane, but as observed by other
researchers, the kinematic wave model can sometimes overestimate the
discharge because of the predicted lack of the point of inflection on
the rising limb of the hydrograph.
2.5.1.2 The Law of Resistance
The DarcyWeisbach friction factor Cf is frequently used in
overland flow studies. This is expressed as
r 8gR'Sf 2%
cf 
 2
U
m
 2
P Um
(2.28)
where R' is the hydraulic radius of the crosssectional area (cross
sectional area of the flow divided by its wetted perimeter and
usually assumed equal to the flow depth, h, of the overland flow).
The friction factor is a function of the flow Reynolds number
0mh
(Ref = ^) and the boundary roughness.
For laminar flows over smooth boundaries the relationship is
(2.29)
56
For laminar flow over rough boundaries Cf can be represented by
(2.30)
where K is a parameter related to the characteristics of the bed
surface and can be as large as 40,000 for dense turf (Wool hi ser,
1975).
For overland flow with rainfall, the raindrop impacts increase
the K factor and it has been represented by the expression
(2.31)
where KQ is the K value without rainfall and ar and bp are empirical
coefficients. Tables with typical values for KQ, ar, and br are pre
sented in Woolhiser's (1975) study. Woolhiser also indicated that
for smooth boundaries (K0 = 24) the raindrop impact effect is
important, but it becomes insignificant for vegetated surfaces (K0
> 3000).
Izzard (1944) was among the first researchers to use this
approach in his study of runoff over rough paved plots. His results
suggested the following equation
= 27(0.21 I4/3 + 1)
(2.32)
Shen and Li (1973), using data from various studies of overland
flow with rainfall over smooth boundaries, proposed the following
equation if Ref < 900.
57
cf
24 + 27.162 I
0.407
Re,
(2.33)
The transition to an apparent turbulent regime has been report
ed at flow Reynolds numbers from 100 to 1000. The higher values usu
ally corresponded to the smooth boundaries. Shen and Li (1973) used
Ref = 900 as the maximum Reynolds number in the laminar flow regime
over smooth surface while Yoon (1970) established this maximum Ref
in the range of 1000.
Savat (1977) has presented a summary of other maximum laminar
Reynolds numbers reported in the literature. He considered that a
turbulent flow was believed to prevail when Ref > 1000, the transi
tional flow occurred when Ref 500 and a laminar flow when Ref < 250.
Savat also indicated that the maximum laminar Reynolds number changed
with changes on the bed slope as seen in other investigations.
For the turbulent flow regime, there are many proposed rela
tionships to use. The Blasius equation
cf =
0.233
Ref0'25
(2.34)
(Woolhiser, 1975) can be used for smooth boundary flows without rain
fall and a Reynolds number less than about 30,000. Robertson et al.
(1966) used the same type of equation to express the friction factors
for three different rough boundaries under rainfall conditions. Un
fortunately, for the flow Reynolds number range tested in the study
(400 < Ref < 4500), the coefficients of their equations changed for
each rough surface studied.
58
Another equation which is frequently used for the turbulent
flow regime is Manning's equation in English units (assuming R' = h)
0 = Li9 s h2/3
m Nm f
(2.35)
Robertson et al. introduced the DarcyWeisbach equation and solved
for Cf to obtain
8g nm2
1.49 h1/6
(2.36)
or
Sfl/10 ... V/5. c
( VRef)!/5 i.499/5 Ref1/5 Ref1/5
(2.37)
In this form the equation has a similar form to Blasius equation
(Equation 2.34). Robertson et al. (1966) reported that in one of
their three rough surfaces studied the value of the flow Reynolds
number exponent was equal to 0.20 = 1/5 with correlation coefficient
of 0.74 for the Ref range between 550 and 4500.
For larger Ref values the effects of viscosity and rainfall
are diminished and the friction factor is usually considered constant
for that bed surface. Consequently, the Chezy's equation can be used
since Cf no longer depends on Ref.
Yen et al. (1972) developed equations for the friction slope
(Sf), the total head slope (Sh) and the dissipated energy gradi
ent (Se) from the general equations presented in Yen and Wenzel
59
(1970) and in Yen (1972) for overland flow with or without rainfall
over smooth boundary case. Then each loss gradient term was written
in DarcyWeisbach resistance coefficient form (Equation 2.28) and
showed that each slope term was numerically different to each other.
The difference in magnitude among the coefficient depended on the
flow conditions (i.e., with or without rainfall or lateral rainfall).
Based on their results, they suggested that for steady spatially var
ied flow computations the momentum equation was preferred to the
energy equation or total head equation, particularly if the Darcy
Weisbach's Cf, Manning's NM, or Chezy's C coefficients are used
as the resistance factors.
Shen and Li (1973) also developed equations for the friction
factor and other parameters for rainfall conditions over "smooth"
surfaces based on the ratios of each parameter value (i.e., water
depth, water discharge, mean velocity, boundary shear stress, Froude
number and friction factor) under rainfall conditions and the equiva
lent parameter without rainfall with the same flow discharge rate.
This is like using the flow Reynolds number (Ref) as the scaling
number. They also obtained Equation 2.33 to calculate the friction
factor in the laminar range (Re < 900). For the very turbulent
regime (Re > 2000), the friction factor was obtained from Blasius'
equation but with a different coefficient value for rainfall condi
tions. For the intermediate flow regime (900 < Ref < 2000) a lin
early interpolated equation was proposed. These previous equations
were obtained by regression analysis of their data and from the lit
erature.
60
Their statistical analysis indicated that the uncertainty in
the selection of the friction factor for the computation of flow
depth and boundary shear stress was not too sensitive and that the
error in using incorrect friction factors was not cummulative with
each step of their numerical model. Their equations were recommended
under the conditions of being used only for 126 < Ref < 12,600,
0.5 in./hr (12.5 mm/hr) < I < 17.5 in./hr (445 mm/hr), 0.005 < S0 <
0.0108 and over a physically "smooth" boundary.
Savat (1977) presented a good summary of the hydraulics of
sheet flow on physically smooth surfaces. He also discussed some
roughness conditions and presented equations for flowmean velocity,
friction factor and Manning's N^. The variation of the exponents
of the water depth and the bed slope terms in the equations due to
the flow regime (i.e., laminar, transition or turbulent) were also
discussed. His comparison with available literature suggested that
sheet flow could be either laminar or purely turbulent, but that
mixed flows prevailed on low slopes (under 5% slopes) combined with
greater depths. He also indicated that in most cases sheet flows
were supercritical, specially on steep slopes.
Savat1s equations and experiments indicated that the effect of
raindrop impacts on the DarcyWeisbach friction factor, Cf, did not
exceed 20% in the case of laminar flow on gentle slopes. He also
indicated that the rainfall influence diminished when the discharge
or the Reynolds number increased as well as when the bed slope angle
increased. Savat also used an equation for Cf in hydraulically
61
smooth turbulent flows, originally presented by Keulegan (1938), in
which Savat rearranged by using the DarcyWeisbach equation (Equation
2.28) and the flow Reynolds number definition to obtain the expres
sion
_L = 5.75 Â£og(Ref(Cf/8)1/2) + constant (2.39)
C ^
cf
Julien and Simons (1985) also suggested the use of the equation
originally proposed by Keulegan (1938), but they used Blasius'
equation for this kind of flow. Their definition for hydraulically
0 5
smooth flow was that the viscous sublayer, 6 = 11.6 v (P/Tq) was
greater than three times the size of the sediment particles, ds.
When the thickness of the viscous sublayer is small compared to
sediment size the flow is considered hvdraulically rough and the
logarithmic equation also given by Keulegan (1938) was considered to
apply. This equation was presented by Julien and Simons as
(BÂ£,1/2
cf
= C = a^ tog(a2
(2.40)
where aj and a2 are constants. However, they used approximated
power relationships such as Manning's equation to express the
friction factor.
Thornes (1980) also presented a similar expression to Equation
2.40 to obtain the friction factor which was originally used by Wol
man (1955).
Savat (1980) considered the resistance to flow in rough super
critical sheet flow which is present on steep slope flows. However,
62
the study only considered overland flow with no rainfall. Expres
sions to obtain the DarcyWeisbach friction factor were obtained for
both laminar and turbulent flow regimes based on the classical Cf
equations and compared with his laboratory results. The observed
values were found to be higher than the ones obtained from classical
equations. The discrepancies were associated with the great varia
tion on the relative depth of standing and travelling waves usually
found on steep slope flows, and due to the turbulence and wake forma
tion around the bottom grains.
2.5.1.3 Boundary Shear Stress
Kinematic Approach. The timemean boundary shear stress t0
is related to the friction slope (Sf) by the equation
T0=yR'Sf (2.41)
For the case of overland flow the hydraulic radius, R1, is
equal to the water depth, h. Some researchers have used the water
depth for their tq calculations. Another frequently used approxi
mation is the assumption of bed slope, S = sine, being equal to the
friction slope based on the relative magnitude of the terms on the
momentum equation (the kinematic wave method). Another reason for
this assumption is the problem of estimating the friction slope
especially under field conditions. Vegetation and cover material
over the soil make it practically impossible to directly measure the
parameters in order to calculate the friction slope.
63
These assumptions lead to the equation
t0 = yti S (2.42)
which some researchers have used as the real value for Tq while
others have used it correctly as a first approximation only.
Dynamic Approach. Another form to obtain t0 is by solving
the dynamic equation (Equation 2.19) for and obtaining from
Equation 2.41. Based on the assumptions made by the authors of each
study, the representation of the dynamic equation may be slightly
different. As mentioned before, Keulegan (1944) was the first to
express that equation for the case of spatially varied flow like the
case of overland flow with rainfall. Other articles which have pre
sented their derivations for this equation or at least have presented
possible methods to solve it are Woo and Brater (1962), Morgali and
Lindsey (1965), Grace and Eagleson (1965, 1966) Ligget and Woolhiser
(1967), AbdelRazaq et al. (1967), Chen and Chow (1968), Chow (1969),
Morgali (1970), Eagleson (1970), Yen and Wenzel (1970), Yen (1972),
Yen et al. (1972), Kilinc and Richardson (1973), and others. The
dynamic equation can only be solved by numerical techniques due to
the complexity of the equation.
Keulegan (1944) recommended that before any approximate solu
tion is attempted the dependence of the friction factor on the flow
Reynolds number is required to be well known. Izzard (1944) was
among the first to present that relationship from curve fitting of
data collected from rougher paved plots. Izzard also obtained that
the water depth was proportional to the cubic root of the
64
longitudinal distance from the upper end of the slope. This rela
tionship had a certain limit which was associated with the change in
the flow regime from laminar to turbulent flow. The onethird power
was also associated with Equation 2.25 used in the kinematic wave
method given that the flow discharge per unit width is expressed as
q = lx.
Yoon (1970) presented in his doctoral dissertation very signi
ficant information about xo over physically smooth surfaces when sim
ulated rainfall was applied to overland flow. His measured values
obtained from a flat surface hotfilm sensor agreed very well with
the computed T0 values from the one dimensional spatially varied flow
equation developed by Yen and Wenzel (1970), Equation 2.20a. This
showed the applicability of the onedimensional dynamic equation of
spatially varied flow for practical purposes.
Yoon indicated that, for a constant flow Reynolds number, tq
increased appreciably with increasing rainfall intensities. This
happened for Refvalues of up to approximately 1000. The rainfall
intensity effect became negligible as Ref further increased. He
also showed that the relationship expressed by Equation 2.30 was val
id for overland flow with rainfall and Ref < 1000 with the constant
K increasing with an increasing rainfall intensity and with a small
dependence of the bed slope. Later, Shen and Li (1973) indicated
that the slope effect was not significant on the Kvalue. So both
studies clearly showed that the flow Reynolds number and rainfall
intensity are the most important parameters affecting the timemean
boundary shear stress.
65
Kisi sel1 s (1971) study showed the same trend as Yoon's results.
The increase in T0 and Cf with the increase in the rainfall intensity
was equally observed over both physically smooth and rough surfaces
and particularly for laminar and transitional flow Reynolds numbers.
In this study the Cf values also increased when the bed slope
increased. For the physically rough surface case the increase in
Cf was slightly larger than that observed for flows with rainfall
over the smooth surface. He indicated that for both surfaces, the
main factor affecting the friction factor values was found to be the
rainfall input.
The studies of Yoon (1970), Kisisel (1971), and Shen and Li
(1973) were conducted at nearly horizontal uniform slopes (S0 < 3%)
with a fixed bed. These are ideal conditions in which the dynamic
equation was found to be a useful tool to evaluate and Cf. Kilinc
and Richardson (1973) also used the dynamic equation for that purpose
too, but their study was conducted at steeper bed slopes {5.7% < SQ <
40%) and with a movable bed (silty sand).
Kilinc and Richardson obtained introducing Equations 2.18
and 2.24 into Equation 2.19. Their study considered steady state
conditions with B = 1, BÂ¡ = 1, q = (I f)x, assumed that the in
filtration rate (f) was constant along the bed slope and used h =
q/um to express the water depth. The solution for T0 at the
downstream end of the plot was obtained numerically using their ex
perimental data. These values which included rainfall effects
were found to be less than the calculated from Shen and Li's ex
pression for Cf, Equation 2.33, but greater than the t calculated
assuming uniform flow, Equation 2.28. The ^values were later
66
used in that study to develop empirical equations for the sediment
discharge of that sloped area subjected to rainfall.
2.5.1.4 Entrainment Motion and Critical Shear Stress
The discrete soil particles of the bed in ar\y stream are
subjected to tractive forces (e.g., shear stress) and lift which try
to move the particles. They are, of course, also subjected to resist
ing forces (e.g., buoyant weight and frictional forces) which will
prevent the particle motion. When the tractive forces are equal to
the resisting forces, the particle will be in an entrainment condi
tion. Under this condition the magnitude of the timemean bed shear
stress (tq) will be considered as the timemean critical shear
stress value (^cr) of the instantaneous tractive force to have been
applied to the bed surface area. This critical value is basically a
function of the particle density, size, shape and roughness, and the
arrangement of the individual particles in the bed surface.
If the acting force is larger than a critical value, the parti
cle will be set in motion. The resulting modes of transport (i.e.,
rolling, saltation or suspension) depend on how much greater the act
ing force is with respect to critical force. If the acting force is
greater but nearly equal to the critical force of the particle roll
ing or sliding will be the predominant mode of transportation. A
greater acting force can make the particle start saltation motion and
when the acting force is high enough, the particle will be suspended
in the stream. So, for a given acting force, higher than the criti
cal one, the larger size particles will usually move by rolling or
67
saltation and the smaller size particles will predominantly move by
suspension.
Usually particles transported in suspension are referred to as
the suspended load. The particles which move by rolling or saltation
are referred to in the literature as bedload. There is also the so
called washload which is made up of grain sizes finer than the bulk
of the bed particles and thus is rarely found in the bed of the
stream. These particles are usually washed through the section of
the stream. Some authors have considered washload and suspended load
as the same load in their sediment transport definition. The total
sediment transport load is referred to as the sum of bedload and
suspended load and/or washload depending on the definition used by
the authors. In this study, the total load will be considered as the
summation of bedload and suspended load with washload being included
into the suspended load.
The critical force required to begin the motion of particles
has been associated basically with two theories. First, Hjulstrorn
(1935; Graf, 1984) presented an erosiondeposition criteria based on
the crosssectional mean flow velocity (0m) required to move parti
cles of certain size. Figure 2.2 shows this basic erosiondeposition
criterion for uniform particles. It shows the limiting zone at which
incipient motion starts and the line of demarcation between the sedi
ment transport and sedimentation. The diagram also indicates that
loose fine sand is the easiest to erode and that the greater resis
tance to erosion in the smaller particle range must depend on the
cohesion forces.
68
Figure 2.2. ErosionDeposition Criteria for Uniform Particles
(after Hjulstrom, 1935; Graf, 1984; reprint with permission of Water
Resources Publications).
500
69
The second theory is based on the timemean critical shear
stress, Tcr. DuBuat (Graf, 1984) used this approach during the late
eighteenth century, but it did not become popular until the beginning
of this century when Schoklitsch published his results (Graf, 1984).
Since then, other researchers have used this approach too.
In 1936, Shields (Graf, 1984) used the shear velocity, v^,
which represents a measure of the intensity of turbulent fluctuations
near the bottom boundary. This is related to the be